\markbothSPECHT \VersionNoIntroduction

This package contains functions for computing the decomposition matrices for Iwahori--Hecke algebras of the symmetric groups. As the (modular) representation theory of these algebras closely resembles that of the (modular) representation theory of the symmetric groups --- indeed, the later is a special case of the former --- many of the combinatorial tools from the representation theory of the symmetric group are included in the package.

These programs grew out of the attempts by Gordon James and myself [JM1]
to understand the decomposition matrices of Hecke algebras of type **A**
when *<q>=-1*. The package is now much more general and its highlights
include:

1. SPECHT provides a means of working in the Grothendieck ring of a
Hecke algebra `H` using the three natural bases corresponding to the
Specht modules, projective indecomposable modules, and simple modules.

2. For Hecke algebras defined over fields of characteristic zero we have implemented the algorithm of Lascoux, Leclerc, and Thibon [LLT] for computing decomposition numbers and ``crystallized decomposition matrices''. In principle, this gives all of the decomposition matrices of Hecke algebras defined over fields of characteristic zero.

3. We provide a way of inducing and restricting modules. In addition,
it is possible to ``induce'' decomposition matrices; this function is
quite effective in calculating the decomposition matrices of Hecke
algebras for small `n`.

4. The `q`--analogue of Schaper's theorem [JM2] is included, as is
Kleshchev's [K] algorithm of calculating the Mullineux map. Both are
used extensively when inducing decomposition matrices.

5. SPECHT can be used to compute the decomposition numbers of
`q`--Schur algebras (and the general linear groups), although there is
less direct support for these algebras. The decomposition matrices for the
`q`--Schur algebras defined over fields of characteristic zero for *n<11*
and all `e` are included in SPECHT.

6. The Littlewood--Richard rule, its inverse, and functions for many of the standard operations on partitions (such as calculating cores, quotients, and adding and removing hooks), are included.

7. The decomposition matrices for the symmetric groups * S_{n}* are
included for

**The modular representation theory of Hecke algebras**

The ``modular'' representation theory of the Iwahori--Hecke algebras of
type **A** was pioneered by Dipper and James [DJ1,DJ2]; here we briefly
outline the theory, referring the reader to the references for details.
The definition of the Hecke algebra can be found in Chapter Iwahori-Hecke
algebras; see also Hecke.

Given a commutative integral domain `R` and a non--zero unit `q` in `R`,
let *<H>= H_{R, q}* be the Hecke algebra of the symmetric group

`S`

(`rad`

`S`

`S`

`D`

`S`

`rad`

`S`

`D`

`R`

is a field.
Given a non--negative integer *i*, let *[i] _{q}=1+q+...+q^{i-1}*. Define

A partition *μ=(μ _{1},μ_{2},...)* is

`D`

(`D`

(`S`

(`S`

(
Given two partitions *μ* and *ν*, where *ν* is `e`--regular, let
*d _{μν}* be the composition multiplicity of

`D`

(`S`

(`H`

. When the rows and columns are ordered in a way compatible with
dominance,
The indecomposable `H`-modules `P`

(` ν`) are indexed by

`P`

(`S`

(`H`

. Similarly,
`D`

(`S`

(

**Two small examples**

Because of the algorithm of [LLT], in principle, all of decomposition matrices for all Hecke algebras defined over fields of characteristic zero are known and available using SPECHT. The algorithm is recursive; however, it is quite quick and, as with a car, you need never look at the engine:

`gap> H:=Specht(4); # e=4, ``R`

a field of characteristic 0
Specht(e=4, S(), P(), D(), Pq())
gap> InducedModule(H.P(12,2));
P(13,2)+P(12,3)+P(12,2,1)+P(10,3,2)+P(9,6)

The [LLT] algorithm was applied 24 times during this calculation.

For Hecke algebras defined over fields of positive characteristic the
major tool provided by SPECHT, apart from the decomposition matrices
contained in the libraries, is a way of ``inducing'' decomposition
matrices. This makes it fairly easy to calculate the associated
decomposition matrices for ``small'' `n`. For example, the SPECHT
libraries contain the decomposition matrices for the symmetric groups
* S_{n}* over fields of characteristic 3 for

gap> H:=Specht(3,3); # e=3,`R`

field of characteristic 3 Specht(e=3, p=3, S(), P(), D()) gap> d:=DecompositionMatrix(H,5); # known forn<2e5 | 1 4,1 | . 1 3,2 | . 1 1 3,1^2 | . . . 1 2^2,1 | 1 . . . 1 2,1^3 | . . . . 1 1^5 | . . 1 . . gap> for n in [6..14] do > d:=InducedDecompositionMatrix(d); SaveDecompositionMatrix(d); > od;

The function `InducedDecompositionMatrix`

contains almost every trick
that I know for computing decomposition matrices (except using the spin
groups). I would be very happy to hear of any improvements.

SPECHT can also be used to calculate the decomposition numbers of the
`q`--Schur algebras; although, as yet, here no additional routines for
calculating the projective indecomposables indexed by `e`--singular
partitions. Such routines will probably be included in a future release,
together with the (conjectural) algorithm [LT] for computing the
decomposition matrices of the `q`--Schur algebras over fields of
characteristic zero.

In the next release of SPECHT, I will also include functions for
computing the decomposition matrices of Hecke algebras of type **B**, and
more generally those of the Ariki--Koike algebras. As with the Hecke
algebra of type **A**, there is an algorithm for computing the decomposition
matrices of these algebras when `R`

is a field of characteristic zero [M].

**Credits**

I would like to thank Gordon James, Johannes Lipp, and Klaus Lux for their comments and suggestions.

If you find SPECHT useful please let me know. I would also appreciate hearing any suggestions, comments, or improvements. In addition, if SPECHT does play a significant role in your research, please send me a copy of the paper(s) and please cite SPECHT in your references.

The lastest version of SPECHT can be obtained from http://maths.usyd.edu.au:8000/u/mathas/specht.

Andrew Mathas\footnoteSupported in part by SERC grant GR/J37690

%
mathas@maths.usyd.edu.au

University of Sydney, 1997.

**References**

[A] S. Ariki,
`On the decomposition numbers of the Hecke algebra of G(m,1,n)`,
J. Math. Kyoto Univ.,

[B] J. Brundan,
`Modular branching rules for quantum GL_{n} and the Hecke algebra
of type A`, Proc. London Math. Soc (3), to appear.

[DJ1] R. Dipper and G. James,
`Representations of Hecke algebras of general linear groups`,
Proc. London Math. Soc. (3), **52** (1986), 20--52.

[DJ2] R. Dipper and G. James,
`Blocks and idempotents of Hecke algebras of general linear groups`,
Proc. London Math. Soc. (3), **54** (1987), 57--82.

[G] M. Geck,
`Brauer trees of Hecke algebras`, Comm. Alg., **20** (1992), 2937--2973.

[Gr] I. Grojnowski,
`Affine Hecke algebras (and affine quantum GL_{n}) at roots of unity`,
IMRN

[J] G. James,
`The decomposition matrices of GL_{n}(q) for n ≤10`,
Proc. London Math. Soc.,

[JK] G. James and A. Kerber,
`The representation theory of the symmetric group`, **16**,
Encyclopedia of Mathematics, Addison--Wesley, Massachusetts (1981).

[JM1] G. James and A. Mathas,
`Hecke algebras of type A at q=-1`, J. Algebra,

[JM2] G. James and A. Mathas,
`A q--analogue of the Jantzen--Schaper Theorem`, Proc. London Math.
Soc. (3),

[K] A. Kleshchev,
`Branching rules for modular representations III`,
J. London Math. Soc., **54**, 1996, 25--38.

[LLT] A. Lascoux, B. Leclerc, and J-Y. Thibon,
`Hecke algebras at roots of unity and crystal bases of quantum
affine algebras`, Comm. Math. Phys., **181** (1996), 205--263.

[LT] B. Leclerc and J-Y. Thibon,
`Canonical bases and q--deformed Fock spaces`, Int. Research Notices

[M] A. Mathas,
`Canonical bases and the decomposition matrices of Ariki--Koike
algebras`, preprint 1996.

- Specht
- Schur
- DecompositionMatrix
- CrystalizedDecompositionMatrix
- DecompositionNumber
- InducedModule
- SInducedModule
- RestrictedModule
- SRestrictedModule
- InducedDecompositionMatrix
- IsNewIndecomposable
- InvertDecompositionMatrix
- AdjustmentMatrix
- SaveDecompositionMatrix
- CalculateDecompositionMatrix
- MatrixDecompositionMatrix
- DecompositionMatrixMatrix
- AddIndecomposable
- RemoveIndecomposable
- MissingIndecomposables
- SimpleDimension
- SpechtDimension
- Schaper
- IsSimpleModule
- MullineuxMap
- MullineuxSymbol
- PartitionMullineuxSymbol
- GoodNodes
- NormalNodes
- GoodNodeSequence
- PartitionGoodNodeSequence
- GoodNodeLatticePath
- LittlewoodRichardsonRule
- InverseLittlewoodRichardsonRule
- EResidueDiagram
- HookLengthDiagram
- RemoveRimHook
- AddRimHook
- ECore
- IsECore
- EQuotient
- CombineEQuotientECore
- EWeight
- ERegularPartitions
- IsERegular
- ConjugatePartition
- PartitionBetaSet
- ETopLadder
- LengthLexicographic
- Lexicographic
- ReverseDominance
- Specialized
- ERegulars
- SplitECores
- Coefficient of Specht module
- InnerProduct
- SpechtPrettyPrint
- SemistandardTableaux
- StandardTableaux
- ConjugateTableau
- ShapeTableau
- TypeTableau

`Specht(`

`e`)

`Specht(`

`e`, `p`)

`Specht(`

`e`, `p`, `val` [,`HeckeRing`])

Let `R` be a field of characteristic 0, `q` a non--zero element of `R`,
and let `e` be the smallest positive integer such that

1+q+...+q^{e-1}=0 |

`Specht(``e`)

allows calculations in the Grothendieck rings of
the Hecke algebras `H`

of type `Hecke`

Hecke.) Below we also
describe how to consider Hecke algebras defined over fields of positive
characteristic.
`Specht`

returns a record which contains, among other things, functions
`S`

, `P`

, and `D`

which correspond to the Specht modules, projective
indecomposable modules, and the simple modules for the family of Hecke
algebras determined by `R` and `q`. SPECHT allows manipulation of
arbitrary linear combinations of these ``modules'', as well as a way
of inducing and restricting them, ``multiplying'' them, and converting
between these three natural bases of the Grothendieck ring. Multiplication
of modules corresponds to taking a tensor product, and then inducing (thus
giving a module for a larger Hecke algebra).

gap> RequirePackage("specht"); H:=Specht(5); Specht(e=5, S(), P(), D(), Pq()) gap> H.D(3,2,1); D(3,2,1) gap> H.S( last ); S(6)-S(4,2)+S(3,2,1) gap> InducedModule(H.P(3,2,1)); P(4,2,1)+P(3,3,1)+P(3,2,2)+2*P(3,2,1,1) gap> H.S(last); S(4,2,1)+S(3,3,1)+S(3,2,2)+2*S(3,2,1,1)+S(2,2,2,1)+S(2,2,1,1,1) gap> H.D(3,1)*H.D(3); D(7)+2*D(6,1)+D(5,2)+D(5,1,1)+2*D(4,3)+D(4,2,1)+D(3,3,1) gap> RestrictedModule(last); 4*D(6)+3*D(5,1)+5*D(4,2)+2*D(4,1,1)+2*D(3,3)+2*D(3,2,1) gap> H.S(last); S(6)+3*S(5,1)+3*S(4,2)+2*S(4,1,1)+2*S(3,3)+2*S(3,2,1) gap> H.P(last); P(6)+3*P(5,1)+2*P(4,2)+2*P(4,1,1)+2*P(3,3)

The way in which the partitions indexing the modules are printed can
be changed using `SpechtPrettyPrint`

SpechtPrettyPrint.

There is also a function `Schur`

Schur for doing calculations with
the `q`--Schur algebra. See `DecompositionMatrix`

DecompositionMatrix,
and `CrystalizedDecompositionMatrix`

CrystalizedDecompositionMatrix.

This function requires the package ``specht'' (see RequirePackage).

**The functions 'H.S', 'H.P', and 'H.D'**

The functions `H.S`

, `H.P`

, and `H.D`

return records which correspond
to Specht modules, projective indecomposable modules, and simple
modules respectively. Each of these three functions can be called in
four different ways, as we now describe.

`H.S`

(` μ`)

`H.P`

(`H.D`

(
In the first form, ` μ` is a partition (either a list, or a sequence of
integers), and the corresponding Specht module, PIM, or simple module
(respectively), is returned.

gap> H.P(4,3,2); P(4,3,2)

`H.S`

(`x`) `H.P`

(`x`) `H.D`

(`x`)

Here, `x` is an `H`--module. In this form, `H.S`

rewrites `x` as a linear
combination of Specht modules, if possible. Similarly, `H.P`

and `H.D`

rewrite `x` as a linear combination of PIMs and simple modules
respectively. These conversions require knowledge of the relevant
decomposition matrix of `H`; if this is not known then `false`

is
returned (over fields of characteristic zero, all of the decomposition
matrices are known via the algorithm of [LLT]; various other
decomposition matrices are included with SPECHT). For example,
`H.S`

(`H.P`

(` μ`)) returns

∑_{ν} d_{νμ} `S` (ν), |

`false`

if some of these decomposition multiplicities are not known.

gap> H.D( H.P(4,3,2) ); D(5,3,1)+2*D(4,3,2)+D(2,2,2,2,1) gap> H.S( H.D( H.S(1,1,1,1,1) ) ); -S(5)+S(4,1)-S(3,1,1)+S(2,1,1,1)

As the last example shows, SPECHT does not always behave as expected.
The reason for this is that Specht modules indexed by `e`--singular
partitions can always be written as a linear combination of Specht
modules which involve only `e`--regular partitions. As such, it is not
always clear when two elements are equal in the Grothendieck ring.
Consequently, to test whether two modules are equal you should first
rewrite both modules in the `D`

--basis; this is `not` done by SPECHT
because it would be very inefficient.

`H.S`

(`d`, ` μ`)

`H.P`

(`H.D`

(
In the third form, `d` is a decomposition matrix and ` μ` is a
partition. This is useful when you are trying to calculate a new
decomposition matrix

`H.P`

and `H.D`

use `P`

(`D`

(`H.S`

uses `S`

(`false`

is returned.

`gap> H:=Specht(3,3); # e = 3, p = 3 = characteristic of ``R`

Specht(e=3, p=3, S(), P(), D())
gap> d:=InducedDecompositionMatrix(DecompositionMatrix(H,14));;
# Inducing....
The following projectives are missing from <d>:
[ 15 ] [ 8, 7 ]
gap> H.P(d,4,3,3,2,2,1);
S(4,3,3,2,2,1)+S(4,3,3,2,1,1,1)+S(4,3,2,2,2,1,1)+S(3,3,3,2,2,1,1)
gap> H.S(d,7, 3, 3, 2);
D(11,2,1,1)+D(10,3,1,1)+D(8,5,1,1)+D(8,3,3,1)+D(7,6,1,1)+D(7,3,3,2)
gap> H.D(d,14,1);
false

The final example returned `false`

because the partitions `(14,1)`

and `(15)`

have the same *3*--core (and `P`

(15) is missing from `d`).

`H.S`

(`d`, `x`) `H.P`

(`d`, `x`) `H.D`

(`d`, `x`)

In the final form, `d` is a decomposition matrix and `x` is a module. All
three functions rewrite `x` in their respective basis using `d`. Again
this is only useful when you are trying to calculate a new decomposition
matrix because, for any ``known'' decomposition matrix `d`, `H.S(`

and `x`)`H.S`

(`d`, `x`) are equivalent (and similarly for `H.P`

and `H.D`

).

gap> H.S(d, H.D(d,10,5) ); -S(13,2)+S(10,5)

The last example looked at Hecke algebras with parameter `q`=1 and `R` a
field of characteristic 3 (so `e`=3); that is, the group algebra of the
symmetric group over a field of characteristic 3. More, generally, the
command `Specht(`

can be used to consider the group algebras of
the symmetric groups over fields of characteristic `p`, `p`)`p` (i.e. `e`=p, and
`R`

a field of characteristic `p`).

For example, the dimensions of the simple modules of * S_{6}*
over fields of characteristic 5 can be computed as follows:

gap> H:=Specht(5,5);; SimpleDimension(H,6); 6 : 1 5,1 : 5 4,2 : 8 4,1^2 : 10 3^2 : 5 3,2,1 : 8 3,1^3 : 10 2^3 : 5 2^2,1^2 : 1 2,1^4 : 5

To consider Hecke algebras defined over arbitrary fields `Specht`

must
also be supplied with a **valuation map** `val` as an argument. The
function `val` is a map from some PID into the natural numbers; at
present it is needed only by functions which rely (at least implicitly),
upon the `q`--analogue of Schaper's theorem. In general, `val` depends
upon `q` and the characteristic of `R`; full details can be found in [JM2].

Over fields of characteristic zero, and in the symmetric group case, the
function `val` is automatically defined by `Specht`

. When `R` is a field
of characteristic zero, `val`(*[i] _{q}*) is

As another example, if *<q>=4* and `R` is a field of characteristic 5
(so *<e>=2*), then the valuation map sends the integer `x` to
*ν _{5}([4]_{x})* where

gap> val:=function(x) local v; > x:=Sum([0..x-1],v->4^v); # x->[x]\_q > v:=0; while x mod 5=0 do x:=x/5; v:=v+1; od; > return v; > end;; gap> H:=Specht(2,5,val,"e2q4"); Specht(e=2, p=5, S(), P(), D(), HeckeRing="e2q4")

Notice the string ``e2q4'' which was also passed to `Specht`

in this
example. Although it is not strictly necessary, it is a good idea when
using a ``non--standard'' valuation map `val` to specify the value
of `H.HeckeRing`

=`HeckeRing`. This string is used for internal
bookkeeping by SPECHT; in particular, it is used to determine filenames
when reading and saving decomposition matrices. If a ``standard''
valuation map is used then `HeckeRing` is set to the string
``*e<e>p<p>*''; otherwise it defaults to ``unknown''. The
function `SaveDecompositionMatrix`

will not save any decomposition
matrix for any Hecke algebra `H`

with `H.HeckeRing`

=``unknown''.

For Hecke algebras `H` defined over fields of characteristic zero Lascoux,
Leclerc and Thibon [LLT] have described an easy, inductive, algorithm for
calculating the decomposition matrices of `H`. Their algorithm really
calculates the **canonical basis**, or (global) **crystal basis** of the
Fock space; results of Grojnowski--Lusztig [Gr] show that computing this
basis is equivalent to computing the decomposition matrices of `H` (see
also [A]).

The **Fock space** *𝔽* is an (integrable) module for the quantum group
*U _{q}(\widehatsl_{e})* of the affine special linear group.

`v`

]--module with basis the set of all Specht modules
`S`

( 𝔽 = ⊕_{n ≥0}⊕_{μ\vdash n}*C*[`v` ] `S` (μ); |

`v`

=`H.info.Indeterminate`

is an indeterminate over the integers
(or strictly, `Pq`

(`Pq`

(`P`

(`H.Pq`

(To access the elements of the Fock space SPECHT provides the functions:

`H.Pq`

(` μ`)

`H.Sq`

(
Notice that, unlike `H.P`

and `H.S`

the only arguments which `H.Pq`

and
`H.Sq`

accept are partitions. (Given that our indeterminate is `v`

these
functions should really be called `H.Pv`

and `H.Sv`

; here ``q'' stands
for ``quantum'```
.)
The function
```

H.Pq` computes the canonical basis element `

Pq`(`

S` μ`)
of the Fock space corresponding to the

`(`*μ*) of *𝔽*.

```
gap> H:=Specht(4);
Specht(e=4, S(), P(), D(), Pq())
gap> H.Pq(6,2);
S(6,2)+v*S(5,3)
gap> RestrictedModule(last);
S(6,1)+(v + v^(-1))*S(5,2)+v*S(4,3)
gap> H.P(last);
P(6,1)+(v + v^(-1))*P(5,2)
gap> Specialized(last);
P(6,1)+2*P(5,2)
gap> H.Sq(5,3,2);
S(5,3,2)
gap> InducedModule(last,0);
v^(-1)*S(5,3,3)
```

```
The modules returned by
```

H.Pq` and `

H.Sq```
behave very much like elements
of the Grothendieck ring of
````H`; however, they should be considered as
elements of the Fock space. The key difference is that when induced or
restricted ``quantum'' analogues of induction and restriction are used.
These analogues correspond to the action of *U*_{q}(\widehatsl_{e})
on *𝔽* [LLT].
In effect, the functions

H.Pq` and `

H.Sq```
allow computations in
the Fock space, using the functions
```

InducedModule```
"InducedModule" and
```

RestrictedModule` "RestrictedModule". The functions `

H.S`, `

H.P```
, and
```

H.D```
can also be applied to elements of the Fock space, in which case
they have the expected effect. In addition, any element of the Fock space
can be specialized to give the corresponding element of the Grothendieck
ring of
```

H```
(it is because of this correspondence that we do not make a
distinction between elements of the Fock space and the Grothendieck
ring of
```

H```
).
When working over fields of characteristic zero \Specht will
automatically calculate any canonical basis elements that it needs for
computations in the Grothendieck ring of
````H`. If you are not interested
in the canonical basis elements you need never work with them directly.
If, for some reason, you do not want \Specht to use the canonical basis
elements to calculate decomposition numbers then all you need to do is

Unbind`(`

H.Pq').
`Schur(`

`e`)

`Schur(`

`e`, `p`)

`Schur(`

`e`, `p`, `val` [,`HeckeRing`])

This function behaves almost identically to the function `Specht`

(see
Specht), the only difference being that the three functions in the
record `S`

returned by `Schur`

are called `S.W`

, `S.P`

, and `S.F`

and that they correspond to the q-Weyl modules, the projective
decomposable modules, and the simple modules of the q--Schur algebra
respectively. Note that our labeling of these modules is non--standard,
following that used by James in [J]. The standard labeling can be
obtained from ours by replacing all partitions by their conjugates.

Almost all of the functions in SPECHT which accept a `Specht`

record `H` will also accept a record `S` returned by `Schur`

In the current version of SPECHT the decomposition matrices of q--Schur
algebras are not fully supported. The `InducedDecompositionMatrix`

function can be applied to these matrices; however there are no additional
routines available for calculating the columns corresponding to
`e`--singular partitions. The decomposition matrices for the q--Schur
algebras defined over a field of characteristic 0 for *<n> ≤10* are in
the SPECHT libraries.

gap> S:=Schur(2); Schur(e=2, W(), P(), F(), Pq()) gap> InducedDecompositionMatrix(DecompositionMatrix(S,3)); # The following projectives are missing fromd: # [ 2, 2 ] 4 | 1 #`DecompositionMatrix`

(S,4) returns the 3,1 | 1 1 # full decomposition matrix. The point 2^2 | . 1 . # of this example is to emphasize the 2,1^2 | 1 1 . 1 # current limitations of`Schur`

. 1^4 | 1 . . 1 1

Note that when `S` is defined over a field of characteristic zero then
it contains a function `S.Pq`

for calculating canonical basis elements
(see `Specht`

Specht); currently `S.Pq(`

is implemented only
for ` μ`)

`H.Wq`

.
See also `Specht`

Specht. This function requires the package
``specht'' (see RequirePackage).

`DecompositionMatrix(`

`H`, `n` [,`Ordering`])

`DecompositionMatrix(`

`H`, `filename` [,`Ordering`])

The function `DecompositionMatrix`

returns the decomposition matrix `D`

of
* H( S_{n})* where

`H`

is a Hecke algebra record returned by the function
`Specht`

(or `Schur`

). `DecompositionMatrix`

first checks to see whether
the required decomposition matrix exists as a library file (checking first
in the current directory, next in the directory specified by
`SpechtDirectory`

, and finally in the SPECHT libraries). If `H.Pq`

exists, `DecompositionMatrix`

next looks for `CrystalizedDecompositionMatrix`

CrystalizedDecompositionMatrix).
If the decomposition matrix `d`

is not stored in the library
`DecompositionMatrix`

will calculate `d`

when `H`

is a Hecke algebra with a
base field `R`

of characteristic zero, and will return `false`

otherwise
(in which case the function `CalculateDecompositionMatrix`

CalculateDecompositionMatrix can be used to force SPECHT to try and
calculate this matrix).
For Hecke algebras defined over fields of characteristic zero, SPECHT
uses the algorithm of [LLT] to calculate decomposition matrices
(this feature can be disabled by unbinding `H.Pq`

). The decomposition
matrices for the `q`--Schur algebras for *<n> ≤10* are contained in the
SPECHT library, as are those for the symmetric group over fields of
positive characteristic when *<n><15*.

Once a decomposition matrix is known, SPECHT keeps an internal copy
of it which is used by the functions `H.S`

, `H.P`

, and `H.D`

; these
functions also read decomposition matrix files as needed.

If you set the variable `SpechtDirectory`

, then SPECHT will also search
for decomposition matrix files in this directory. The files in the current
directory override those in `SpechtDirectory`

and those in the SPECHT
libraries.

In the second form of the function, when a `filename` is supplied,
`DecompositionMatrix`

will read the decomposition matrix in the file
`filename`, and this matrix will become SPECHT's internal copy of
this matrix.

By default, the rows and columns of the decomposition matrices are ordered
lexicographically. This can be changed by supplying `DecompositionMatrix`

with an ordering function such as `LengthLexicographic`

or
`ReverseDominance`

. You do not need to specify the ordering you want
every time you call `DecompositionMatrix`

; SPECHT will keep the same
ordering until you change it again. This ordering can also be set ``by
hand'' using the variable `H.Ordering`

.

gap> DecompositionMatrix(Specht(3),6,LengthLexicographic); 6 | 1 5,1 | 1 1 4,2 | . . 1 3^2 | . 1 . 1 4,1^2 | . 1 . . 1 3,2,1 | 1 1 . 1 1 1 2^3 | 1 . . . . 1 3,1^3 | . . . . 1 1 2^2,1^2| . . . . . . 1 2,1^4 | . . . 1 . 1 . 1^6 | . . . 1 . . .

Once you have a decomposition matrix it is often nice to be able
to print it. The on screen version is often good enough; there is also
a `TeX`

command which generates a **LaTeX** version. There are also
functions for converting SPECHT decomposition matrices into **GAP3**
matrices and visa versa (see `MatrixDecompositionMatrix`

MatrixDecompositionMatrix and `DecompositionMatrixMatrix`

DecompositionMatrixMatrix).

Using the function `InducedDecompositionMatrix`

(see
InducedDecompositionMatrix), it is possible to induce a decomposition
matrix. See also `SaveDecompositionMatrix`

SaveDecompositionMatrix and
`IsNewIndecomposable`

IsNewIndecomposable, `Specht`

Specht, `Schur`

Schur, and `CrystalizedDecompositionMatrix`

CrystalizedDecompositionMatrix.
This function requires the package ``specht'' (see RequirePackage).

`CrystalizedDecompositionMatrix(`

`H`, `n` [,`Ordering`])

`CrystalizedDecompositionMatrix(`

`H`, `filename` [,`Ordering`])

This function is similar to `DecompositionMatrix`

, except that it
returns a **crystallized decomposition matrix**. The columns of decomposition
matrices correspond to projective indecomposables; the columns of
crystallized decomposition matrices correspond to the canonical basis
elements of the Fock space (see Specht). Consequently, the entries in
these matrices are polynomials (in `v`

), and by specializing (i.e. setting
`v`

equal to *1*; see Specialized), the decomposition matrices of `H`
are obtained (see Specht).

Crystallized decomposition matrices are defined only for Hecke algebras over a base field of characteristic zero. Unlike ``normal'' decomposition matrices, crystallized decomposition matrices cannot be induced.

gap> CrystalizedDecompositionMatrix(Specht(3), 6); 6 | 1 5,1 | v 1 4,2 | . . 1 4,1^2 | . v . 1 3^2 | . v . . 1 3,2,1 | v v^2 . v v 1 3,1^3 | . . . v^2 . v 2^3 | v^2 . . . . v 2^2,1^2| . . . . . . 1 2,1^4 | . . . . v v^2 . 1^6 | . . . . v^2 . . gap> Specialized(last); # set`v`

equal to1. 6 | 1 5,1 | 1 1 4,2 | . . 1 4,1^2 | . 1 . 1 3^2 | . 1 . . 1 3,2,1 | 1 1 . 1 1 1 3,1^3 | . . . 1 . 1 2^3 | 1 . . . . 1 2^2,1^2| . . . . . . 1 2,1^4 | . . . . 1 1 . 1^6 | . . . . 1 . .

See also `Specht`

Specht, `Schur`

Schur, `DecompositionMatrix`

DecompositionMatrix, and `Specialized`

Specialized. This function
requires the package ``specht'' (see RequirePackage).

`DecompositionNumber(`

`H`, ` μ`,

`DecompositionNumber(``d`, *μ*, *ν*)

This function attempts to calculate the decomposition multiplicity of
`D`

(` ν`) in

`S`

(`S`

(`P`

(`P`

(`DecompositionNumber`

tries to calculate the answer
using ``row and column removal'' (see [J,Theorem 6.18]).

gap> H:=Specht(6);; gap> DecompositionNumber(H,[6,4,2],[6,6]); 0

This function requires the package ``specht'' (see RequirePackage).

`InducedModule(`

`x`)

`InducedModule(`

`x`, *r _{1}* [,

There is an natural embedding of * H( S_{n})* in

`H`

(`H`

(`H`

(`InducedModule`

returns the induced modules of the Specht
modules, principal indecomposable modules, and simple modules (more
accurately, their image in the Grothendieck ring).
There is also a function `SInducedModule`

(see SInducedModule) which
provides a much faster way of `r`--inducing `s` times (and inducing `s`
times).

Let ` μ` be a partition. Then the induced module

`InducedModule(S(`*μ*))

is easy to describe: it has the same
composition factors as
`S`

(

gap> H:=Specht(2,2); Specht(e=2, p=2, S(), P(), D()) gap> InducedModule(H.S(7,4,3,1)); S(8,4,3,1)+S(7,5,3,1)+S(7,4,4,1)+S(7,4,3,2)+S(7,4,3,1,1) gap> InducedModule(H.P(5,3,1)); P(6,3,1)+2*P(5,4,1)+P(5,3,2) gap> InducedModule(H.D(11,2,1)); # D(x), unable to rewritexas a sum of simples S(12,2,1)+S(11,3,1)+S(11,2,2)+S(11,2,1,1)

When inducing indecomposable modules and simple modules, `InducedModule`

first rewrites these modules as a linear combination of Specht modules
(using known decomposition matrices), and then induces this linear
combination of Specht modules. If possible SPECHT then rewrites the
induced module back in the original basis. Note that in the last example
above, the decomposition matrix for * S_{15}* is not known by SPECHT;
this is why

`InducedModule`

was unable to rewrite this module in the
`D`

--basis.

`InducedModule`

(`x`, *r _{1}* [,

Two Specht modules `S`

(` μ`) and

`S`

(`S`

(`S`

(`InducedModule(S(`*τ*))

, for some partition
`S`

(`S`

(`InducedModule`

allows one to induce ``within blocks'' by only adding
nodes of some fixed

gap> H:=Specht(4); InducedModule(H.S(5,2,1)); S(6,2,1)+S(5,3,1)+S(5,2,2)+S(5,2,1,1) gap> InducedModule(H.S(5,2,1),0); 0*S() gap> InducedModule(H.S(5,2,1),1); S(6,2,1)+S(5,3,1)+S(5,2,1,1) gap> InducedModule(H.S(5,2,1),2); 0*S() gap> InducedModule(H.S(5,2,1),3); S(5,2,2)

The function `EResidueDiagram`

(EResidueDiagram), prints the diagram
of ` μ`, labeling each node with its

gap> EResidueDiagram(H,5,2,1); 0 1 2 3 0 3 0 2

**``Quantized\'\'\ induction**

When `InducedModule`

is applied to the canonical basis elements
`H.Pq`

(` μ`) (or more generally elements of the Fock space; see
Specht), a ``quantum analogue'' of induction is applied. More
precisely, the function

`InducedModule(*,i)`

corresponds to the
action of the generator

gap> H:=Specht(3);; InducedModule(H.Pq(4,2),1,2); S(6,2)+v*S(4,4)+v^2*S(4,2,2) gap> H.P(last); P(6,2)

See also `SInducedModule`

SInducedModule, `RestrictedModule`

RestrictedModule, and `SRestrictedModule`

SRestrictedModule. This
function requires the package ``specht'' (see RequirePackage).

`SInducedModule(`

`x`, `s`)

`SInducedModule(`

`x`, `s`, `r`)

The function `SInducedModule`

, standing for ``string induction'',
provides a more efficient way of `r`--inducing `s` times (and a way of
inducing `s` times if the residue `r` is omitted); `r`--induction is
explained in InducedModule.

gap> H:=Specht(4);; SInducedModule(H.P(5,2,1),3); P(8,2,1)+3*P(7,3,1)+2*P(7,2,2)+6*P(6,3,2)+6*P(6,3,1,1)+3*P(6,2,1,1,1) +2*P(5,3,3)+P(5,2,2,1,1) gap> SInducedModule(H.P(5,2,1),3,1); P(6,3,1,1) gap> InducedModule(H.P(5,2,1),1,1,1); 6*P(6,3,1,1)

Note that the multiplicity of each summand of `InducedModule(x,r,...,r)`

is divisible by *<s>!* and that `SInducedModule`

divides by this constant.

As with `InducedModule`

this function can also be applied to elements of
the Fock space (see Specht), in which case the quantum analogue of
induction is used.

See also `InducedModule`

InducedModule. This function requires the
package ``specht'' (see RequirePackage).

`RestrictedModule(`

`x`)

`RestrictedModule(`

`x`, ` r_{1}` [,

Given a module `x` for `H`

(**S**_{n})`RestrictedModule`

returns the
corresponding module for * H( S_{n-1})*. The restriction of the Specht
module

`S`

(`S`

(`S`

(
There is also a function `SRestrictedModule`

(see SRestrictedModule)
which provides a faster way of `r`--restricting `s` times (and restricting
`s` times).

When more than one residue if given to `RestrictedModule`

it returns

`RestrictedModule` (x,r_{1},r_{2},...,r_{k})=
`RestrictedModule` (`RestrictedModule` (x,r_{1}),r_{2},...,r_{k}) |

`InducedModule`

InducedModule).

gap> H:=Specht(6);; RestrictedModule(H.P(5,3,2,1),4); 2*P(4,3,2,1) gap> RestrictedModule(H.D(5,3,2),1); D(5,2,2)

**``Quantized\'\'\ restriction**

As with `InducedModule`

, when `RestrictedModule`

is applied to the
canonical basis elements `H.Pq`

(` μ`) a quantum analogue of restriction
is applied; this time,

`RestrictedModule(*,i)`

corresponds to the action
of the generator
See also `InducedModule`

InducedModule, `SInducedModule`

SInducedModule, and `SRestrictedModule`

SRestrictedModule. This
function requires the package ``specht''
(see RequirePackage).

`SRestrictedModule(`

`x`, `s`)

`SRestrictedModule(`

`x`, `s`, `r`)

As with `SInducedModule`

this function provides a more efficient way of
`r`--restricting `s` times, or restricting `s` times if the residue `r`
is omitted (cf. `SInducedModule`

SInducedModule).

gap> H:=Specht(6);; SRestrictedModule(H.S(4,3,2),3); 3*S(4,2)+2*S(4,1,1)+3*S(3,3)+6*S(3,2,1)+2*S(2,2,2) gap> SRestrictedModule(H.P(5,4,1),2,4); P(4,4)

See also `InducedModule`

InducedModule, `SInducedModule`

SInducedModule, and `RestrictedModule`

RestrictedModule. This function
requires the package ``specht'' (see RequirePackage).

`InducedDecompositionMatrix(`

`d`)

If `d` is the decomposition matrix of * H( S_{n})*, then

`InducedDecompositionMatrix(``d`)

attempts to calculate the decomposition
matrix of `H`

(`H`

(`InducedDecompositionMatrix`

then tries
to decompose these projectives using the function `IsNewIndecomposable`

(see IsNewIndecomposable). In general there will be columns of the
decomposition matrix which `InducedDecompositionMatrix`

is unable to
decompose and these will have to be calculated ``by hand''.
`InducedDecompositionMatrix`

prints a list of those columns of the
decomposition matrix which it is unable to calculate (this list is also
printed by the function `MissingIndecomposables(``d`)

).

gap> d:=DecompositionMatrix(Specht(3,3),14);; gap> InducedDecompositionMatrix(d);; # Inducing.... The following projectives are missing from <d>: [ 15 ] [ 8, 7 ]

Note that the missing indecomposables come in ``pairs'' which map to
each other under the Mullineux map (see `MullineuxMap`

MullineuxMap).

Almost all of the decomposition matrices included in SPECHT were
calculated directly by `InducedDecompositionMatrix`

. When `n` is
``small'' `InducedDecompositionMatrix`

is usually able to return the
full decomposition matrix for * H( S_{n+1})*.

Finally, although the `InducedDecompositionMatrix`

can also be applied to
the decomposition matrices of the `q`--Schur algebras (see `Schur`

Schur), `InducedDecompositionMatrix`

is much less successful in inducing
these decomposition matrices because it contains no special routines for
dealing with the indecomposable modules of the `q`--Schur algebra which
are indexed by `e`--singular partitions. Note also that we use a
non--standard labeling of the decomposition matrices of `q`--Schur
algebras; see Schur.

`IsNewIndecomposable(`

`d`, `x` [,` μ`])

`IsNewIndecomposable`

is the function which does all of the hard work when
the function `InducedDecompositionMatrix`

is applied to decomposition
matrices (see InducedDecompositionMatrix). Given a projective module
`x`, `IsNewIndecomposable`

returns `true`

if it is able to show that `x`
is indecomposable (and this indecomposable is not already listed in `d`),
and `false`

otherwise. `IsNewIndecomposable`

will also print a brief
description of its findings, giving an upper and lower bound on the
**first** decomposition number ` μ` for which it is unable to determine
the multiplicity of

`S`

(
`IsNewIndecomposable`

works by running through all of the partitions
` ν` such that

`P`

`Schaper`

Schaper),
the Mullineux map (see `MullineuxMap`

MullineuxMap), and inducing simple
modules, to determine if `P`

(`H`

(`IsNewIndecomposable`

will probably use some of the decomposition matrices
of `H`

(
For example, in calculating the 2--modular decomposition matrices of
* S_{r}* the first projective which

`InducedDecompositionMatrix`

is
unable to calculate is `P`

(10).

gap> H:=Specht(2,2);; gap> d:=InducedDecompositionMatrix(DecompositionMatrix(H,9));; # Inducing. # The following projectives are missing fromd: # [ 10 ]

(In fact, given the above commands, SPECHT will return the full
decomposition matrix for * S_{10}* because this matrix is in the library;
these were the commands that I used to calculate the decomposition matrix
in the library.)

By inducing `P`

(9) we can find a projective `H`--module which contains
`P`

(10). We can then use `IsNewIndecomposable`

to try and decompose this
induced module into a sum of PIMs.

gap> SpechtPrettyPrint(); x:=InducedModule(H.P(9),1); S(10)+S(9,1)+S(8,2)+2S(8,1^2)+S(7,3)+2S(7,1^3)+3S(6,3,1)+3S(6,2^2) +4S(6,2,1^2)+2S(6,1^4)+4S(5,3,2)+5S(5,3,1^2)+5S(5,2^2,1)+2S(5,1^5) +2S(4^2,2)+2S(4^2,1^2)+2S(4,3^2)+5S(4,3,1^3)+2S(4,2^3)+5S(4,2^2,1^2) +4S(4,2,1^4)+2S(4,1^6)+2S(3^3,1)+2S(3^2,2^2)+4S(3^2,2,1^2) +3S(3^2,1^4)+3S(3,2^2,1^3)+2S(3,1^7)+S(2^3,1^4)+S(2^2,1^6)+S(2,1^8) +S(1^10) gap> IsNewIndecomposable(d,x); # The multiplicity of S(6,3,1) in P(10) is at least 1 and at most 2. false gap> x; S(10)+S(9,1)+S(8,2)+2S(8,1^2)+S(7,3)+2S(7,1^3)+2S(6,3,1)+2S(6,2^2) +3S(6,2,1^2)+2S(6,1^4)+3S(5,3,2)+4S(5,3,1^2)+4S(5,2^2,1)+2S(5,1^5) +2S(4^2,2)+2S(4^2,1^2)+2S(4,3^2)+4S(4,3,1^3)+2S(4,2^3)+4S(4,2^2,1^2) +3S(4,2,1^4)+2S(4,1^6)+2S(3^3,1)+2S(3^2,2^2)+3S(3^2,2,1^2) +2S(3^2,1^4)+2S(3,2^2,1^3)+2S(3,1^7)+S(2^3,1^4)+S(2^2,1^6)+S(2,1^8) +S(1^10)

Notice that some of the coefficients of the Specht modules in `x` have
changed; this is because `IsNewIndecomposable`

was able to determine
that the multiplicity of `S`

(6,3,1) was at most *2* and so it
subtracted one copy of `P`

(6,3,1) from `x`.

In this case, the multiplicity of `S`

(6,3,1) in `P`

(10) is easy to resolve
because general theory says that this multiplicity must be odd. Therefore,
*x- P(6,3,1)* is projective. After subtracting

`P`

(6,3,1) from `IsNewIndecomposable`

to see if `IsNewIndecomposable`

that all of the multiplicities up to
and including `S`

(6,3,1) have already been checked by giving it the
addition argument

gap> x:=x-H.P(d,6,3,1);; IsNewIndecomposable(d,x,6,3,1); true

Consequently, *<x>= P(10)* and we add it to the decomposition matrix

`gap> AddIndecomposable(d,x); SaveDecompositionMatrix(d); `

A full description of what `IsNewIndecomposable`

does can be found by
reading the comments in `specht.g`

. Any suggestions or improvements on
this function would be especially welcome.

See also `DecompositionMatrix`

DecompositionMatrix and
`InducedDecompositionMatrix`

InducedDecompositionMatrix. This function
requires the package ``specht'' (see RequirePackage).

`InvertDecompositionMatrix(d)`

Returns the inverse of the (`e`--regular part of) `d`, where `d` is a
decomposition matrix, or crystallized decomposition matrix, of a Hecke
algebra or `q`--Schur algebra. If part of the decomposition matrix `d`
is unknown then `InvertDecompositionMatrix`

will invert as much of `d`
as possible.

gap> H:=Specht(4);; d:=CrystalizedDecompositionMatrix(H,5);; gap> InvertDecompositionMatrix(d); 5 | 1 4,1 | . 1 3,2 | -v . 1 3,1^2| . . . 1 2^2,1| v^2 . -v . 1 2,1^3| . . . . . 1

See also `DecompositionMatrix`

DecompositionMatrix, and
`CrystalizedDecompositionMatrix`

CrystalizedDecompositionMatrix. This
function requires the package ``specht'' (see RequirePackage).

`AdjustmentMatrix(`

`dp`, `d`)

James [J] noticed, and Geck [G] proved, that the decomposition
matrices
`dp` for Hecke algebras defined over fields of positive characteristic
admit a factorization

dp = d * a |

gap> H:=Specht(2);; Hp:=Specht(2,2);; gap> d:=DecompositionMatrix(H,13);; dp:=DecompositionMatrix(Hp,13);; gap> a:=AdjustmentMatrix(dp,d); 13 | 1 12,1 | . 1 11,2 | 1 . 1 10,3 | . . . 1 10,2,1 | . . . . 1 9,4 | 1 . 1 . . 1 9,3,1 | 2 . . . . . 1 8,5 | . 1 . . . . . 1 8,4,1 | 1 . . . . . . . 1 8,3,2 | . 2 . . . . . 1 . 1 7,6 | 1 . . . . 1 . . . . 1 7,5,1 | . . . . . . 1 . . . . 1 7,4,2 | 1 . 1 . . 1 . . . . 1 . 1 7,3,2,1| . . . . . . . . . . . . . 1 6,5,2 | . 1 . . . . . 1 . 1 . . . . 1 6,4,3 | 2 . . . 1 . . . . . . . . . . 1 6,4,2,1| . 2 . 1 . . . . . . . . . . . . 1 5,4,3,1| 4 . 2 . . . . . . . . . . . . . . 1 gap> MatrixDecompositionMatrix(dp)= > MatrixDecompositionMatrix(d)*MatrixDecompositionMatrix(a); true

In the last line we have checked our calculation.

See also `DecompositionMatrix`

DecompositionMatrix, and
`CrystalizedDecompositionMatrix`

CrystalizedDecompositionMatrix. This
function requires the package ``specht'' (see RequirePackage).

`SaveDecompositionMatrix(`

`d`)

`SaveDecompositionMatrix(`

`d`, `filename`)

The function `SaveDecompositionMatrix`

saves the decomposition matrix `d`.
After a decomposition matrix has been saved, the functions `H.S`

,
`H.P`

, and `H.D`

will automatically access it as needed. So, for example,
before saving `d` in order to retrieve the indecomposable `P`

(` μ`)
from

`H.P(``d`, *μ*)

; once `H.P(`*μ*)

suffices.
Since `InducedDecompositionMatrix(`

consults the decomposition
matrices for smaller `d`)`n`, if they are available, it is advantageous to
save decomposition matrices as they are calculated. For example, over a
field of characteristic *5*, the decomposition matrices for the symmetric
groups * S_{n}* with

gap> H:=Specht(5,5);; gap> d:=DecompositionMatrix(H,9);; gap> for r in [10..20] do > d:=InducedDecompositionMatrix(d); > SaveDecompositionMatrix(d); > od;

If your Hecke algebra record `H`

is defined using a non--standard
valuation map (see Specht) then it is also necessary to set the string
```H.HeckeRing`

'', or to supply the function with a `filename` before
it will save your matrix. `SaveDecompositionMatrix`

will also save
adjustment matrices and the various other matrices that appear in SPECHT
(they can be read back in using `DecompositionMatrix`

). Each matrix
has a default filename which you can over ride by supplying a `filename`.
Using non--standard file names will stop SPECHT from automatically
accessing these matrices in future.

See also DecompositionMatrix `DecompositionMatrix`

DecompositionMatrix and `CrystalizedDecompositionMatrix`

CrystalizedDecompositionMatrix. This function requires the package
``specht'' (see RequirePackage).

`CalculateDecompositionMatrix(H,n)`

`CalculateDecompositionMatrix(H,n)`

is similar to the function
`DecompositionMatrix`

DecompositionMatrix in that both functions try to
return the decomposition matrix `d`

of * H( S_{n})*; the difference is
that this function tries to calculate this matrix whereas the later reads
the matrix from the library files (in characteristic zero both functions
apply the algorithm of [LLT] to compute

`d`

). In effect this function is
only needed when working with Hecke algebras defined over fields of positive
characteristic (or when you wish to avoid the libraries).
For example, if you want to do calculations with the decomposition matrix of
the symmetric group * S_{15}* over a field of characteristic two,

`DecompositionMatrix`

returns false whereas `CalculateDecompositionMatrix`

;
returns a part of the decomposition matrix.

gap> H:=Specht(2,2); Specht(e=2, p=2, S(), P(), D()) gap> d:=DecompositionMatrix(H,15); # This decomposition matrix is not known; use CalculateDecompositionMatrix() # or InducedDecompositionMatrix() to calculate with this matrix. false gap> d:=CalculateDecompositionMatrix(H,15);; # Projective indecomposable P(6,4,3,2) not known. # Projective indecomposable P(6,5,3,1) not known. ... gap> MissingIndecomposables(d); The following projectives are missing from <d>: [ 15 ] [ 14, 1 ] [ 13, 2 ] [ 12, 3 ] [ 12, 2, 1 ] [ 11, 4 ] [ 11, 3, 1 ] [ 10, 5 ] [ 10, 4, 1 ] [ 10, 3, 2 ] [ 9, 6 ] [ 9, 5, 1 ] [ 9, 4, 2 ] [ 9, 3, 2, 1 ] [ 8, 7 ] [ 8, 6, 1 ] [ 8, 5, 2 ] [ 8, 4, 3] [ 8, 4, 2, 1 ] [ 7, 6, 2 ] [ 7, 5, 3 ] [ 7, 5, 2, 1 ] [ 7, 4, 3, 1 ] [ 6, 5, 4 ] [ 6, 5, 3, 1 ] [ 6, 4, 3, 2 ]

Actually, you are much better starting with the decomposition matrix of
* S_{14}* and then applying

`InducedDecompositionMatrix`

to this matrix.
See also DecompositionMatrix `DecompositionMatrix`

. This function
requires the package ``specht'' (see RequirePackage).

`MatrixDecompositionMatrix(`

`d`)

Returns the **GAP3** matrix corresponding to the SPECHT decomposition
matrix `d`. The rows and columns of `d` are ordered by `H.Ordering`

.

gap> MatrixDecompositionMatrix(DecompositionMatrix(Specht(3),5)); [ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ], [ 1, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ] ]

See also `DecompositionMatrix`

DecompositionMatrix and
`DecompositionMatrixMatrix`

DecompositionMatrixMatrix. This function
requires the package ``specht'' (see RequirePackage).

`DecompositionMatrixMatrix(`

`H`, `m`, `n`)

Given a Hecke algebra `H`, a **GAP3** matrix `m`, and an integer `n` this
function returns the SPECHT decomposition matrix corresponding to `m`.
If `p`

is the number of partitions of `n` and `r`

the number of
`e`--regular partitions of `n`, then `m` must be either *<r>× r*,

`H.Ordering`

(see Specht).

gap> H:=Specht(3);; gap> m:=[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 1, 0 ], > [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ];; gap> DecompositionMatrixMatrix(H,m,4); 4 | 1 3,1 | . 1 2^2 | 1 . 1 2,1^2| . . . 1 1^4 | . . 1 .

See also `DecompositionMatrix`

DecompositionMatrix and
`MatrixDecompositionMatrix`

MatrixDecompositionMatrix. This function
requires the package ``specht'' (see RequirePackage).

`AddIndecomposable(`

`d`, `x`)

`AddIndecomposable(`

inserts the indecomposable module `d`, `x`)`x` into
the decomposition matrix `d`. If `d` already contains the indecomposable
`x` then a warning is printed. The function `AddIndecomposable`

also
calculates `MullineuxMap(`

(see MullineuxMap) and adds this
indecomposable to `x`)`d` (or checks to see that it agrees with the
corresponding entry of `d` if this indecomposable is already by `d`).

See `IsNewIndecomposable`

IsNewIndecomposable for an example.
See also `DecompositionMatrix`

DecompositionMatrix and
`CrystalizedDecompositionMatrix`

CrystalizedDecompositionMatrix. This
function requires the package ``specht'' (see RequirePackage).

`RemoveIndecomposable(`

`d`, ` μ`)

The function `RemoveIndecomposable`

removes the column from `d` which
corresponds to `P`

(` μ`). This is sometimes useful when trying to
calculate a new decomposition matrix using SPECHT and want to test a
possible candidate for a yet to be identified PIM.

See also `DecompositionMatrix`

DecompositionMatrix and
`CrystalizedDecompositionMatrix`

CrystalizedDecompositionMatrix. This
function requires the package ``specht'' (see RequirePackage).

`MissingIndecomposables(`

`d`)

The function `MissingIndecomposables`

prints the list of partitions
corresponding to the indecomposable modules which are not listed in `d`.

See also `DecompositionMatrix`

DecompositionMatrix and
`CrystalizedDecompositionMatrix`

CrystalizedDecompositionMatrix. This
function requires the package ``specht'' (see RequirePackage).

`SimpleDimension(`

`d`)

`SimpleDimension(`

`H`, `n`)

`SimpleDimension(`

`H`\|`d`, ` μ`)

In the first two forms, `SimpleDimension`

prints the dimensions of all
of the simple modules specified by `d` or for the Hecke algebra
* H( S_{n})* respectively. If a partition

`D`

gap> H:=Specht(6);; gap> SimpleDimension(H,11,3); 272 gap> d:=DecompositionMatrix(H,5);; SimpleDimension(d,3,2); 5 gap> SimpleDimension(d); 5 : 1 4,1 : 4 3,2 : 5 3,1^2 : 6 2^2,1 : 5 2,1^3 : 4 1^5 : 1

This function requires the package ``specht'' (see RequirePackage).

`SpechtDimension(`

` μ`)

Calculates the dimension of the Specht module `S`

(` μ`), which is equal
to the number of standard

gap> SpechtDimension(6,3,2,1); 5632

See also `SimpleDimension`

SimpleDimension. This function requires
the package ``specht'' (see RequirePackage).

`Schaper(`

`H`, ` μ`)

Given a partition ` μ`, and a Hecke algebra

`Schaper`

returns a
linear combination of Specht modules which have the same composition
factors as the sum of the modules in the ``Jantzen filtration'' of
`S`

(`D`

(`S`

(`Schaper(`*μ*)

.
`Schaper`

uses the valuation map `H.valuation`

attached to `H` (see
Specht and [JM2]).

One way in which the `q`--Schaper theorem can be applied is as follows.
Suppose that we have a projective module `x`, written as a linear
combination of Specht modules, and suppose that we are trying to decide
whether the projective indecomposable `P`

(` μ`) is a direct summand of

`P`

(`P`

(`InnerProduct(Schaper(H,`*μ*),x)

is non--zero (note, in particular, that
we don't need to know the indecomposable `P`

(
The `q`--Schaper theorem can also be used to check for irreduciblity; in
fact, this is the basis for the criterion employed by `IsSimpleModule`

.

gap> H:=Specht(2);; gap> Schaper(H,9,5,3,2,1); S(17,2,1)-S(15,2,1,1,1)+S(13,2,2,2,1)-S(11,3,3,2,1)+S(10,4,3,2,1)-S(9,8,3) -S(9,8,1,1,1)+S(9,6,3,2)+S(9,6,3,1,1)+S(9,6,2,2,1) gap> Schaper(H,9,6,5,2); 0*S(0)

The last calculation shows that `S`

(9,6,5,2) is irreducible when `R` is a
field of characteristic *0* and

(cf. `e`=2`IsSimpleModule(H,9,6,5,2)`

).

This function requires the package ``specht'' (see RequirePackage).

`IsSimpleModule(`

`H`, ` μ`)

` μ` an

Given an `e`--regular partition ` μ`,

`IsSimpleModule(``H`, *μ*)

returns `true`

if `S`

(`false`

otherwise. This
calculation uses the valuation function `H.valuation`

; see Specht. Note
that the criterion used by `IsSimpleModule`

is completely combinatorial;
it is derived from the

gap> H:=Specht(3);; gap> IsSimpleModule(H,45,31,24); false

See also `Schaper`

Schaper. This function requires the package
``specht'' (see RequirePackage).

`MullineuxMap(`

`e`\|`H`, ` μ`)

`MullineuxMap(``d`, *μ*)

`MullineuxMap(``x`)

Given an integer `e`, or a SPECHT record `H`, and a partition ` μ`,

`MullineuxMap`

(
The sign representation `D`

(*1 ^{n}*) of the Hecke algebra is the (one
dimensional) representation sending

`D`

(`D`

(`D`

(`D`

(
Deep results of Kleshchev [K] for the symmetric group give another
(proven) algorithm for calculating the partition *μ ^{#} * (Ford and
Kleshchev have deduced Mullineux's conjecture from this). Using the
canonical basis, it was shown by [LLT] that the natural generalization
of Kleshchev's algorithm to

Kleshchev's map is easy to describe; he proved that if `gns` is any
good node sequence for ` μ`, then the sequence obtained from

`GoodNodeSequence`

GoodNodeSequence).

gap> MullineuxMap(Specht(2),12,5,2); [ 12, 5, 2 ] gap> MullineuxMap(Specht(4),12,5,2); [ 4, 4, 4, 2, 2, 1, 1, 1 ] gap> MullineuxMap(Specht(6),12,5,2); [ 4, 3, 2, 2, 2, 2, 2, 1, 1 ] gap> MullineuxMap(Specht(8),12,5,2); [ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 ] gap> MullineuxMap(Specht(10),12,5,2); [ 3, 3, 3, 3, 2, 1, 1, 1, 1, 1 ]

`MullineuxMap`

(`d`, ` μ`)

The Mullineux map can also be calculated using a decomposition matrix.
To see this recall that ``tensoring'' a Specht module `S`

(` μ`) with
the sign representation yields a module isomorphic to the dual of

`S`

(`MullineuxMap`

uses
`MullineuxMap`

(`x`)

In the third form, `x` is a module, and `MullineuxMap`

returns *<x> ^{#} *, the
image of

`P`

(`P`

(`InducedDecompositionMatrix`

).
See also `GoodNodes`

GoodNodes and `GoodNodeSequence`

GoodNodeSequence . This function requires the package ``specht''
(see RequirePackage).

`MullineuxSymbol(`

`e`\|`H`, ` μ`)

Returns the Mullineux symbol of the `e`--regular partition ` μ`.

gap> MullineuxSymbol(5,[8,6,5,5]); [ [ 10, 6, 5, 3 ], [ 4, 4, 3, 2 ] ]

See also `PartitionMullineuxSymbol`

PartitionMullineuxSymbol. This function
requires the package ``specht'' (see RequirePackage).

`PartitionMullineuxSymbol(`

`e`\|`H`, `ms`)

Given a Mullineux symbol `ms`, this function returns the corresponding
`e`--regular partition.

gap> PartitionMullineuxSymbol(5, MullineuxSymbol(5,[8,6,5,5]) ); [ 8, 6, 5, 5 ]

See also `MullineuxSymbol`

MullineuxSymbol. This function
requires the package ``specht'' (see RequirePackage).

`GoodNodes(`

`e`\|`H`, ` μ`)

`GoodNodes(``e`\|`H`, *μ*, `r`)

Given a partition and an integer `e`, Kleshchev [K] defined the notion of
**good node** for each residue `r` (*0 ≤ r<e*). When `e` is prime and
` μ` is

`D`

(
By definition, there is at most one good node for each residue `r`, and
this node is a removable node (in the diagram of ` μ`). The function

`GoodNodes`

returns a list of the rows of `false`

if there
is no good node of residue

gap> GoodNodes(5,[5,4,3,2]); [ false, false, 2, false, 1 ] gap> GoodNodes(5,[5,4,3,2],0); false gap> GoodNodes(5,[5,4,3,2],4); 1

The good nodes also determine the Kleshchev--Mullineux map (see
`GoodNodeSequence`

GoodNodeSequence and `MullineuxMap`

MullineuxMap). This
function requires the package ``specht'' (see RequirePackage).

`NormalNodes(`

`e`\|`H`, ` μ`)

`NormalNodes(``e`\|`H`, *μ*, `r`)

Returns the numbers of the rows of ` μ` which end in one of
Kleshchev's [K] normal nodes. In the second form, only those rows
corresponding to normal nodes of the specified residue are returned.

gap> NormalNodes(5,[6,5,4,4,3,2,1,1,1]); [ [ 1, 4 ], [ ], [ ], [ 2, 5 ], [ ] ] gap> NormalNodes(5,[6,5,4,4,3,2,1,1,1],0); [ 1, 4 ]

See also `GoodNodes`

GoodNodes. This function requires the package
``specht'' (see RequirePackage).

`GoodNodeSequence(`

`e`\|`H`, ` μ`)

`GoodNodeSequences(``e`\|`H`, *μ*)

` μ` an

Given an `e`--regular partition ` μ` of

`GoodNodes`

GoodNodes). In general, `PartitionGoodNodeSequence`

PartitionGoodNodeSequence).

gap> H:=Specht(4);; GoodNodeSequence(H,4,3,1); [ 0, 3, 1, 0, 2, 2, 1, 3 ] gap> GoodNodeSequence(H,4,3,2); [ 0, 3, 1, 0, 2, 2, 1, 3, 3 ] gap> GoodNodeSequence(H,4,4,2); [ 0, 3, 1, 0, 2, 2, 1, 3, 3, 2 ] gap> GoodNodeSequence(H,5,4,2); [ 0, 3, 1, 0, 2, 2, 1, 3, 3, 2, 0 ]

The function `GoodNodeSequences`

returns the list of all good node
sequences for ` μ`.

gap> GoodNodeSequences(H,5,2,1); [ [ 0, 1, 2, 3, 3, 2, 0, 0 ], [ 0, 3, 1, 2, 2, 3, 0, 0 ], [ 0, 1, 3, 2, 2, 3, 0, 0 ], [ 0, 1, 2, 3, 3, 0, 2, 0 ], [ 0, 1, 2, 3, 0, 3, 2, 0 ], [ 0, 1, 2, 3, 3, 0, 0, 2 ], [ 0, 1, 2, 3, 0, 3, 0, 2 ] ]

The good node sequences determine the Mullineux map (see `GoodNodes`

GoodNodes and `MullineuxMap`

MullineuxMap). This function requires the
package ``specht'' (see RequirePackage).

`PartitionGoodNodeSequence(`

`e`\|`H`, `gns`)

Given a good node sequence `gns` (see `GoodNodeSequence`

GoodNodeSequence), this function returns the unique `e`--regular
partition corresponding to `gns` (or `false`

if in fact `gns` is not a
good node sequence).

gap> H:=Specht(4);; gap> PartitionGoodNodeSequence(H,0, 3, 1, 0, 2, 2, 1, 3, 3, 2); [ 4, 4, 2 ]

See also `GoodNodes`

GoodNodes, `GoodNodeSequence`

GoodNodeSequence
and `MullineuxMap`

MullineuxMap. This function requires the package
``specht'' (see RequirePackage).

`GoodNodeLatticePath(`

`e`\|`H`, ` μ`)

`GoodNodeLatticePaths(``e`\|`H`, *μ*)

`LatticePathGoodNodeSequence(``e`\|`H`, `gns`)

The function `GoodNodeLatticePath`

returns a sequence of partitions which
give a path in the `e`--good partition lattice from the empty partition
to ` μ`. The second function returns the list of all paths in the

gap> GoodNodeLatticePath(3,3,2,1); [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 1, 1 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ] gap> GoodNodeLatticePaths(3,3,2,1); [ [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 1, 1 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ], [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 2 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ] ] gap> GoodNodeSequence(4,6,3,2); [ 0, 3, 1, 0, 2, 2, 3, 3, 0, 1, 1 ] gap> LatticePathGoodNodeSequence(4,last); [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 2 ], [ 3, 2 ], [ 3, 2, 1 ], [ 4, 2, 1 ], [ 4, 2, 2 ], [ 5, 2, 2 ], [ 6, 2, 2 ], [ 6, 3, 2 ] ]

See also `GoodNodes`

GoodNodes. This function requires the package
``specht'' (see RequirePackage).

`LittlewoodRichardsonRule(`

` μ`,

`LittlewoodRichardsonCoefficient(`*μ*, *ν*, *τ*)

Given partitions ` μ` of

`S`

(`S`

(`H`

(`H`

(∑_{ν} a_{μν}^{λ} `S` (λ), |

The function `LittlewoodRichardsonRule`

returns an (unordered) list of
partitions of *n+m* in which each partition ` λ` occurs

`S(`*μ*)*S(*ν*)

.

gap> H:=Specht(0);; # the generic Hecke algebra with`R`

=C[`q`

] gap> LittlewoodRichardsonRule([3,2,1],[4,2]); [ [ 4, 3, 2, 2, 1 ],[ 4, 3, 3, 1, 1 ],[ 4, 3, 3, 2 ],[ 4, 4, 2, 1, 1 ], [ 4, 4, 2, 2 ],[ 4, 4, 3, 1 ],[ 5, 2, 2, 2, 1 ],[ 5, 3, 2, 1, 1 ], [ 5, 3, 2, 2 ],[ 5, 4, 2, 1 ],[ 5, 3, 2, 1, 1 ],[ 5, 3, 3, 1 ], [ 5, 4, 1, 1, 1 ],[ 5, 4, 2, 1 ],[ 5, 5, 1, 1 ],[ 5, 3, 2, 2 ], [ 5, 3, 3, 1 ],[ 5, 4, 2, 1 ],[ 5, 4, 3 ],[ 5, 5, 2 ],[ 6, 2, 2, 1, 1], [ 6, 3, 1, 1, 1 ],[ 6, 3, 2, 1 ],[ 6, 4, 1, 1 ],[ 6, 2, 2, 2 ], [ 6, 3, 2, 1 ],[ 6, 4, 2 ],[ 6, 3, 2, 1 ],[ 6, 3, 3 ],[ 6, 4, 1, 1 ], [ 6, 4, 2 ], [ 6, 5, 1 ], [ 7, 2, 2, 1 ], [ 7, 3, 1, 1 ], [ 7, 3, 2 ], [ 7, 4, 1 ] ] gap> H.S(3,2,1)*H.S(4,2); S(7,4,1)+S(7,3,2)+S(7,3,1,1)+S(7,2,2,1)+S(6,5,1)+2*S(6,4,2)+2*S(6,4,1,1) +S(6,3,3)+3*S(6,3,2,1)+S(6,3,1,1,1)+S(6,2,2,2)+S(6,2,2,1,1)+S(5,5,2) +S(5,5,1,1)+S(5,4,3)+3*S(5,4,2,1)+S(5,4,1,1,1)+2*S(5,3,3,1)+2*S(5,3,2,2) +2*S(5,3,2,1,1)+S(5,2,2,2,1)+S(4,4,3,1)+S(4,4,2,2)+S(4,4,2,1,1)+S(4,3,3,2) +S(4,3,3,1,1)+S(4,3,2,2,1) gap> LittlewoodRichardsonCoefficient([3,2,1],[4,2],[5,4,2,1]); 3

The function `LittlewoodRichardsonCoefficient`

returns a single
Littlewood--Richardson coefficient (although you are really better off
asking for all of them, since they will all be calculated anyway).

See also `InducedModule`

InducedModule and
`InverseLittlewoodRichardsonRule`

InverseLittlewoodRichardsonRule.
This function requires the package ``specht'' (see RequirePackage).

`InverseLittlewoodRichardsonRule(`

` τ`)

Returns a list of all pairs of partitions [*μ,ν*] such that the
Littlewood-Richardson coefficient *a _{μν}^{τ}* is non-zero
(see LittlewoodRichardsonRule). The list returned is unordered and
[

gap> InverseLittlewoodRichardsonRule([3,2,1]); [ [ [ ],[ 3, 2, 1 ] ],[ [ 1 ],[ 3, 2 ] ],[ [ 1 ],[ 2, 2, 1 ] ], [ [ 1 ],[ 3, 1, 1 ] ],[ [ 1, 1 ],[ 2, 2 ] ],[ [ 1, 1 ],[ 3, 1 ] ], [ [ 1, 1 ],[ 2, 1, 1 ] ],[ [ 1, 1, 1 ],[ 2, 1 ] ],[ [ 2 ],[ 2, 2 ] ], [ [ 2 ],[ 3, 1 ] ],[ [ 2 ],[ 2, 1, 1 ] ],[ [ 2, 1 ],[ 3 ] ], [ [ 2, 1 ],[ 2, 1 ] ],[ [ 2, 1 ],[ 2, 1 ] ],[ [ 2, 1 ],[ 1, 1, 1 ] ], [ [ 2, 1, 1 ],[ 2 ] ],[ [ 2, 1, 1 ],[ 1, 1 ] ],[ [ 2, 2 ],[ 2 ] ], [ [ 2, 2 ],[ 1, 1 ] ],[ [ 2, 2, 1 ],[ 1 ] ],[ [ 3 ],[ 2, 1 ] ], [ [ 3, 1 ],[ 2 ] ],[ [ 3, 1 ],[ 1, 1 ] ],[ [ 3, 1, 1 ],[ 1 ] ], [ [ 3, 2 ],[ 1 ] ],[ [ 3, 2, 1 ],[ ] ] ]

See also `LittlewoodRichardsonRule`

LittlewoodRichardsonRule.

This function requires the package ``specht'' (see RequirePackage).

`EResidueDiagram(`

`H`\|`e`, ` μ`)

`EResidueDiagram(``x`)

The `e`--residue of the *(i,j)*--th node in the diagram of a partition
` μ` is

`EResidueDiagram(``e`, *μ*)

prints the
diagram of the partition
If `x` is a module then `EResidueDiagram(`

prints the `x`)`e`--residue
diagrams of all of the `e`--regular partitions appearing in `x` (such
diagrams are useful when trying to decide how to restrict and induce
modules and also in applying results such as the ``Scattering theorem''
of [JM1]). It is not necessary to supply the integer `e` in this case
because `x` ``knows'' the value of `e`.

gap> H:=Specht(2);; EResidueDiagram(H.S(H.P(7,5))); [ 7, 5 ] 0 1 0 1 0 1 0 1 0 1 0 1 [ 6, 5, 1 ] 0 1 0 1 0 1 1 0 1 0 1 0 [ 5, 4, 2, 1 ] 0 1 0 1 0 1 0 1 0 0 1 1 # There are 3 2-regular partitions.

This function requires the package ``specht'' (see RequirePackage).

`HookLengthDiagram(`

` μ`)

Prints the diagram of ` μ`, replacing each node with its hook length
(see [JK]).

gap> HookLengthDiagram(11,6,3,2); 14 13 11 9 8 7 5 4 3 2 1 8 7 5 3 2 1 4 3 1 2 1

This function requires the package ``specht'' (see RequirePackage).

`RemoveRimHook(`

` μ`,

Returns the partition obtained from *μ* by removing the (`row`, `col`)--th
rim hook from (the diagram of) ` μ`.

gap> RemoveRimHook([6,5,4],1,2); [ 4, 3, 1 ] gap> RemoveRimHook([6,5,4],2,3); [ 6, 3, 2 ] gap> HookLengthDiagram(6,5,4); 8 7 6 5 3 1 6 5 4 3 1 4 3 2 1

See also `AddRimHook`

AddRimHook. This function requires the package
``specht'' (see RequirePackage).

`AddRimHook(`

` μ`,

Returns a list [` ν`,

`false`

is returned.

gap> AddRimHook([6,4,3],1,3); [ [ 9, 4, 3 ], 0 ] gap> AddRimHook([6,4,3],2,3); false gap> AddRimHook([6,4,3],3,3); [ [ 6, 5, 5 ], 1 ] gap> AddRimHook([6,4,3],4,3); [ [ 6, 4, 3, 3 ], 0 ] gap> AddRimHook([6,4,3],5,3); false

See also `RemoveRimHook`

RemoveRimHook. This function requires the
package ``specht'' (see RequirePackage).

`ECore(`

`H`\|`e`, ` μ`)

The `e`-core of a partition ` μ` is what remains after as many rim

`ECore(`*μ*)

returns the

gap> H:=Specht(6);; ECore(H,16,8,6,5,3,1); [ 4, 3, 1, 1 ]

The `e`--core is calculated here using James' notation of an **abacus**;
there is also an `EAbacus `

function; but it is more ``pretty'' than
useful.

See also `IsECore`

IsECore, `EQuotient`

EQuotient, and `EWeight`

EWeight. This function requires the package ``specht'' (see
RequirePackage).

`IsECore(`

`H`\|`e`, ` μ`)

Returns `true`

if ` μ` is an

`false`

otherwise; see
`ECore`

ECore.
See also `ECore`

ECore. This function requires the package ``specht''
(see RequirePackage).

`EQuotient(`

`H`\|`e`, ` μ`)

Returns the `e`-quotient of ` μ`; this is a sequence of

gap> H:=Specht(8);; EQuotient(H,22,18,16,12,12,1,1); [ [ 1, 1 ], [ ], [ ], [ ], [ ], [ 2, 2 ], [ ], [ 1 ] ]

See also `ECore`

ECore and `CombineEQuotientECore`

CombineEQuotientECore.
This function requires the package ``specht'' (see RequirePackage).

`CombineEQuotientECore(`

`H`\|`e`, `Q`, `C`)

A partition is uniquely determined by its `e`-quotient and its `e`-core
(see EQuotient and ECore). `CombineEQuotientECore(`

returns
the partition which has `e`, `Q`, `C`)`e`--quotient `Q` and `e`--core `C`. The integer
`e` can be replaced with a record `H` which was created using the function
`Specht`

.

gap> H:=Specht(11);; mu:=[100,98,57,43,12,1];; gap> Q:=EQuotient(H,mu); [ [ 9 ], [ ], [ ], [ ], [ ], [ ], [ 3 ], [ 1 ], [ 9 ], [ ], [ 5 ] ] gap> C:=ECore(H,mu); [ 7, 2, 2, 1, 1, 1 ] gap> CombineEQuotientECore(H,Q,C); [ 100, 98, 57, 43, 12, 1 ]

See also `ECore`

ECore and `EQuotient`

EQuotient. This function
requires the package ``specht'' (see RequirePackage).

`EWeight(`

`H`\|`e`, ` μ`)

The `e`--weight of a partition is the number of `e`--hooks which must be
removed from the partition to reach the `e`--core (see `ECore`

ECore).

gap> EWeight(6,[16,8,6,5,3,1]); 5

This function requires the package ``specht'' (see RequirePackage).

`ERegularPartitions(`

`H`\|`e`, `n`)

A partition *μ=(μ _{1},μ_{2},...)* is

`ERegularPartitions(``e`, `n`)

returns the list of

gap> H:=Specht(3); Specht(e=3, S(), P(), D(), Pq()); gap> ERegularPartitions(H,6); [ [ 2, 2, 1, 1 ], [ 3, 2, 1 ], [ 3, 3 ], [ 4, 1, 1 ], [ 4, 2 ], [ 5, 1 ], [ 6 ] ]

This function requires the package ``specht'' (see RequirePackage).

`IsERegular(`

`H`\|`e`, ` μ`)

Returns `true`

if ` μ` is

`false`

otherwise.
This functions requires the package ``specht'' (see RequirePackage).

`ConjugatePartition(`

` μ`)

Given a partition ` μ`,

`ConjugatePartition(`*μ*)

returns the
partition whose diagram is obtained by interchanging the rows and columns
in the diagram of

gap> ConjugatePartition(6,4,3,2); [ 4, 4, 3, 2, 1, 1 ]

This function requires the package ``specht'' (see RequirePackage).

`PartitionBetaSet(`

`bn`)

Given a **set** of beta numbers `bn` (see `BetaSet`

BetaSet), this function
returns the corresponding partition. Note in particular that `bn` must be
a set of integers.

gap> PartitionBetaSet([ 2, 3, 6, 8 ]); [ 5, 4, 2, 2 ]

This function requires the package ``specht'' (see RequirePackage).

`ETopLadder(`

`H`\|`e`, ` μ`)

The ladders in the diagram of a partition are the lines connecting nodes
of constant `e`--residue, having slope *<e>-1* (see [JK]). A new partition
can be obtained from ` μ` by sliding all nodes up to the highest
possible rungs on their ladders.

`ETopLadder(``e`, *μ*)

returns the
partition obtained in this way; it is automatically

gap> H:=Specht(4);; gap> ETopLadder(H,1,1,1,1,1,1,1,1,1,1); [ 4, 3, 3 ] gap> ETopLadder(6,1,1,1,1,1,1,1,1,1,1); [ 2, 2, 2, 2, 2 ]

This function requires the package ``specht'' (see RequirePackage).

`LengthLexicographic(`

` μ`,

`LengthLexicographic`

returns `true`

if the length of ` μ` is less
than the length of

`Lexicographic(`*μ*, *ν*)

.

gap> p:=Partitions(6);;Sort(p,LengthLexicographic); p; [ [ 6 ],[ 5, 1 ],[ 4, 2 ],[ 3, 3 ],[ 4, 1, 1 ],[ 3, 2, 1 ],[ 2, 2, 2 ], [ 3, 1, 1, 1 ],[ 2, 2, 1, 1 ],[ 2, 1, 1, 1, 1 ],[ 1, 1, 1, 1, 1, 1 ] ]

This function requires the package ``specht'' (see RequirePackage).

`Lexicographic(`

` μ`,

`Lexicographic(`

returns ` μ`,

`true`

if

gap> p:=Partitions(6);;Sort(p,Lexicographic); p; [ [ 6 ],[ 5, 1 ],[ 4, 2 ],[ 4, 1, 1 ],[ 3, 3 ],[ 3, 2, 1 ], [ 3, 1, 1, 1 ],[ 2, 2, 2 ],[ 2, 2, 1, 1 ],[ 2, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1 ] ]

This function requires the package ``specht'' (see RequirePackage).

`ReverseDominance(`

` μ`,

This is another total order on partitions which extends the dominance
ordering (see Dominates). Here ` μ` is greater than

∑_{j ≥ i}μ_{j} > ∑_{j ≥ i}ν_{j}. |

gap> p:=Partitions(6);;Sort(p,ReverseDominance); p; [ [ 6 ], [ 5, 1 ], [ 4, 2 ], [ 3, 3 ], [ 4, 1, 1 ], [ 3, 2, 1 ], [ 2, 2, 2 ], [ 3, 1, 1, 1 ], [ 2, 2, 1, 1 ], [ 2, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1 ] ]

This is the ordering used by James in the appendix of his Springer lecture notes book.

This function requires the package ``specht'' (see RequirePackage).

`Specialized(x [,q]);`

`Specialized(d [,q]);`

Given an element of the Fock space `x` (see Specht), or a crystallized
decomposition matrix (see CrystalizedDecompositionMatrix), `Specialized`

returns the corresponding element of the Grothendieck ring or the
corresponding decomposition matrix of the Hecke algebra respectively. By
default the indeterminate `v`

is specialized to *1*; however `v`

can be
specialized to any (integer) `q` by supplying a second argument.

gap> H:=Specht(2);; x:=H.Pq(6,2); S(6,2)+v*S(6,1,1)+v*S(5,3)+v^2*S(5,1,1,1)+v*S(4,3,1)+v^2*S(4,2,2) +(v^3 + v)*S(4,2,1,1)+v^2*S(4,1,1,1,1)+v^2*S(3,3,1,1)+v^3*S(3,2,2,1) +v^3*S(3,1,1,1,1,1)+v^3*S(2,2,2,1,1)+v^4*S(2,2,1,1,1,1) gap> Specialized(x); S(6,2)+S(6,1,1)+S(5,3)+S(5,1,1,1)+S(4,3,1)+S(4,2,2) +2*S(4,2,1,1)+S(4,1,1,1,1)+S(3,3,1,1)+S(3,2,2,1)+S(3,1,1,1,1,1) +S(2,2,2,1,1)+S(2,2,1,1,1,1) gap> Specialized(x,2); S(6,2)+2*S(6,1,1)+2*S(5,3)+4*S(5,1,1,1)+2*S(4,3,1)+4*S(4,2,2)+10*S(4,2,1,1) +4*S(4,1,1,1,1)+4*S(3,3,1,1)+8*S(3,2,2,1)+8*S(3,1,1,1,1,1)+8*S(2,2,2,1,1) +16*S(2,2,1,1,1,1)

An example of `Specialize`

being applied to a crystallized decomposition
matrix can be found in CrystalizedDecompositionMatrix. This function
requires the package ``specht'' (see RequirePackage).

`ERegulars(`

`x`)

`ERegulars(`

`d`)

`ListERegulars(`

`x`)

`ERegulars(`

prints a list of the `x`)`e`--regular partitions, together
with multiplicities, which occur in the module `x`. `ListERegulars(`

returns an actual list of these partitions rather than printing them.
`x`)

gap> H:=Specht(8);; gap> x:=H.S(InducedModule(H.P(8,5,3)) ); S(9,5,3)+S(8,6,3)+S(8,5,4)+S(8,5,3,1)+S(6,5,3,3)+S(5,5,4,3)+S(5,5,3,3,1) gap> ERegulars(x); [ 9, 5, 3 ] [ 8, 6, 3 ] [ 8, 5, 4 ] [ 8, 5, 3, 1 ] [ 6, 5, 3, 3 ] [ 5, 5, 4, 3 ] [ 5, 5, 3, 3, 1 ] gap> H.P(x); P(9,5,3)+P(8,6,3)+P(8,5,4)+P(8,5,3,1)

This example shows why these functions are useful: given a projective
module `x`, as above, and the list of `e`--regular partitions in
`x` we know the possible indecomposable direct summands of `x`.

Note that it is not necessary to specify what `e` is when calling this
function because `x` ``knows'' the value of `e`.

The function `ERegulars`

can also be applied to a decomposition
matrix `d`; in this case it returns the unitriangular submatrix of `d`
whose rows and columns are indexed by the `e`--regular partitions.

These function requires the package ``specht'' (see RequirePackage).

`SplitECores(`

`x`)

`SplitECores(`

`x`, ` μ`)

`SplitECores(``x`, `y`)

The function `SplitECores(`

returns a list `x`)`[`

where
the Specht modules in each *b _{1}*,...,

`SplitECores(``x`, *μ*)

returns
the component of `SplitECores(``x`, `y`)

returns the component of

gap> H:=Specht(2);; gap> SplitECores(InducedModule(H.S(5,3,1))); [ S(6,3,1)+S(5,3,2)+S(5,3,1,1), S(5,4,1) ] gap> InducedModule(H.S(5,3,1),0); S(5,4,1) gap> InducedModule(H.S(5,3,1),1); S(6,3,1)+S(5,3,2)+S(5,3,1,1)

See also `ECore`

ECore, `InducedModule`

InducedModule, and `RestrictedModule`

RestrictedModule.

This function requires the package ``specht'' (see RequirePackage).

`Coefficient(`

`x`, ` μ`)

If `x` is a sum of Specht (resp. simple, or indecomposable) modules, then
`Coefficient(`

returns the coefficient of `x`, ` μ`)

`S`

(`D`

(`P`

(

gap> H:=Specht(3);; x:=H.S(H.P(7,3)); S(7,3)+S(7,2,1)+S(6,2,1^2)+S(5^2)+S(5,2^2,1)+S(4^2,1^2)+S(4,3^2)+S(4,3,2,1) gap> Coefficient(x,5,2,2,1); 1

This function requires the package ``specht'' (see RequirePackage).

`InnerProduct(`

`x`, `y`)

Here `x` and `y` are some modules of the Hecke algebra (i.e. Specht
modules, PIMS, or simple modules). `InnerProduct(`

computes the
standard inner product of these elements. This is sometimes a convenient
way to compute decomposition numbers (for example).
`x`, `y`)

gap> InnerProduct(H.S(2,2,2,1), H.P(4,3)); 1 gap> DecompositionNumber(H,[2,2,2,1],[4,3]); 1

This function requires the package ``specht'' (see RequirePackage).

`SpechtPrettyPrint(true)`

`SpechtPrettyPrint(false)`

`SpechtPrettyPrint()`

This function changes the way in which SPECHT prints modules. The first two forms turn pretty printing on and off respectively (by default it is off), and the third form toggles the printing format.

gap> H:=Specht(2);; x:=H.S(H.P(6));; gap> SpechtPrettyPrint(true); x; S(6)+S(5,1)+S(4,1^2)+S(3,1^3)+S(2,1^4)+S(1^6) gap> SpechtPrettyPrint(false); x; S(6)+S(5,1)+S(4,1,1)+S(3,1,1,1)+S(2,1,1,1,1)+S(1,1,1,1,1,1) gap> SpechtPrettyPrint(); x; S(6)+S(5,1)+S(4,1^2)+S(3,1^3)+S(2,1^4)+S(1^6)

This function requires the package ``specht'' (see RequirePackage).

`SemistandardTableaux(`

` μ`,

` μ` a partition,

Returns a list of the semistandard ` μ`--tableaux of type

gap> SemistandardTableaux([4,3],[1,1,1,2,2]); [ [ [ 1, 2, 3, 4 ], [ 4, 5, 5 ] ], [ [ 1, 2, 3, 5 ], [ 4, 4, 5 ] ], [ [ 1, 2, 4, 4 ], [ 3, 5, 5 ] ], [ [ 1, 2, 4, 5 ], [ 3, 4, 5 ] ], [ [ 1, 3, 4, 4 ], [ 2, 5, 5 ] ], [ [ 1, 3, 4, 5 ], [ 2, 4, 5 ] ] ]

See also `StandardTableaux`

StandardTableaux. This function requires the
package ``specht'' (see RequirePackage).

`StandardTableaux(`

` μ`)

` μ` a partition.

Returns a list of the standard ` μ`--tableaux.

gap> StandardTableaux(4,2); [ [ [ 1, 2, 3, 4 ], [ 5, 6 ] ], [ [ 1, 2, 3, 5 ], [ 4, 6 ] ], [ [ 1, 2, 3, 6 ], [ 4, 5 ] ], [ [ 1, 2, 4, 5 ], [ 3, 6 ] ], [ [ 1, 2, 4, 6 ], [ 3, 5 ] ], [ [ 1, 2, 5, 6 ], [ 3, 4 ] ], [ [ 1, 3, 4, 5 ], [ 2, 6 ] ], [ [ 1, 3, 4, 6 ], [ 2, 5 ] ], [ [ 1, 3, 5, 6 ], [ 2, 4 ] ] ]

See also `SemistandardTableaux`

SemistandardTableaux. This function requires
the package ``specht'' (see RequirePackage).

`ConjugateTableau(`

`tab`)

Returns the tableau obtained from `tab` by interchangings its rows and
columns.

gap> ConjugateTableau([ [ 1, 3, 5, 6 ], [ 2, 4 ] ]); [ [ 1, 2 ], [ 3, 4 ], [ 5 ], [ 6 ] ]

This function requires the package ``specht'' (see RequirePackage).

`ShapeTableau(`

`tab`)

Given a tableau `tab` this function returns the partition (or composition).

gap> ShapeTableau( [ [ 1, 1, 2, 3 ], [ 4, 5 ] ] ); [ 4, 2 ]

This function requires the package ``specht'' (see RequirePackage).

`TypeTableau(`

`tab`)

Returns the type of the (semistandard) tableau `tab`; that is, the
composition *σ=(σ _{1},σ_{2},...)* where

gap> List(SemistandardTableaux([5,4,2],[4,3,0,1,3]),TypeTableau); [ [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ] ]

This function requires the package ``specht'' (see RequirePackage).

gap3-jm

11 Mar 2019