Self-Mullineux-SUBS

The file selfmuBG.g is a sequence of functions written in GAP 3 to compute the bijection EBeta which takes a prime (or odd integer) e and a self-Mullineux partition and returns the (unique) BG-partition associated under the correspondence defined in this paper.

To use this functions open gap3 in the folder containing selfmuBG.g and do

Read("selfmuBG.g");

Then, any of the functions below can be used.

Warning

The algorithms defining these functions are not the most efficient for computing the combinatorial objects associated.

Functions

• DiagramPartition(mu): Takes a partition mu and returns the set of nodes that form the Young diagram of mu.

• Rim(mu): Takes a partition mu and returns the set of nodes (i,j) of the rim of mu.

• OrgRim(mu): Takes a partition mu and returns a list of the nodes of the rim of the diagram of mu, organized from "north-east" to "south-west" of the Young diagram.

• HookLength(la,i,j): Takes a partition la and a node (i,j) in the Young diagram of la and returns the hook-length of the (i,j)-th hook.

• PartitionDiag(nodes): Takes a set of nodes forming a Young diagram and returns the associated partition.

• DiagBoxRim(mu): Takes a self-conjugate partition mu an returns i the only diagonal where (i,i) is the only diagonal node in the rim of mu.

• PRim1(e,mu): Set U of the p-rim* of mu.

• PRim2(l,e): Fix the middle segment.

• PRim(e,mu): The subset U of the p-rim*.

• PRim4(e,mu): The reflection of subset U of the p-rim*

• PRim5(e,mu): p-rim* for self-conjugate mu.

• ERim(e,mu): Takes an odd prime e and a self-conjugate partition mu and returns [mu^(1)* ,e-rim*] where e-rim is the e-rim* of a self-conjugate partition and mu^(1)* is the partition obtained from mu after deleting the e-rim*.

• AutoSymb(e,mu): takes a prime e and a self-conjugate partition mu and returns the first line of the p-BG-symbol (first line) for mu.

• EBGsymb(p,mu): Takes a prime p and a self-conjugate partition mu and returns the p-BG-symbol mu.

• EBetaInverse(e,la): Takes an odd prime e, and a e-BG-partition la and returns the e-self-Mullineux partition corresponding to la under the bijection defined by the bg-symbol.

• DurfeeNumber(mu): takes a partition and returns its Durfee number.

• EBGPartitions(p,n): Takes an odd prime p and an integer n and returns the list of pBG-partitions of n.

• EBetaSymbol(e,symb): Takes a prime e and the Mullineux symbol symb (list of rows) of a e-self-Mullineux partition and returns the associated e-BG-partition.

• EBeta(e,mu): Takes a prime e and a e-self-Mullineux mu and returns the e-BG-partition partition corresponding to mu under the bijection defined by the bg-symbol.

SUBS

(SUBS stands for stable unitriangular basic sets. See this preprint for more on SUBS.)

The following is a list of functions to test which blocks of the symmetric group have compatible/completely admissible transversals.

• BlockPartitionsCore(e,n,core): Takes an integer e<n and n and returns all partitions in the block of S_n corresponding to the core core.

• BlockPartitionsWeightCore(e,w,core): Same as BlockPartitionsCore, but with the weight w.

• BlockPartitionsCoreERegulars(e,n,core): Returns all e-regular partitions on a block with a given core.

• BlockPartitionsWeightCoreERegulars(e,w,core): Same but with weight.

• BlockPartitionsWeightCoreERegularsSM(e,w,core): Self-Mullineux partitions in the e-block with weight.

• EWeightWPart(e,w,n): returns all e-regular partitions of integers less than n of e weight w.

• ListECores(e,m): Returns the list of e-cores of rank up to m.

• ListECoresSC(e,m): Returns the list of self-conjugate e-cores of rank up to m.

• ECoresList(e,n): list of all the e-cores of partitions of n.

• PartitionsByBlocks(e,n) : list of all partitions of n organised by blocks (corresponding to e-cores).

• SSet(e,w,core): Returns the list formed by partitions with e-core core and e-weight w.

• SSetReg(e,w,core): Returns the list of e-regular partitions in SSet(e,w,core).

• AdmTransversals(e,w,core): Returns a list of all the possible admissible transversals for the e-block of weight w of the block corresponding to the core core.

• Test1AdmTransversals(e,m): For testing if the above function gives in fact all admissible transversals we have to look for an example where there exists at least two adm. tr. For this, the following function looks for a block where this happens. The test function will look for e-blocks where the rank of the e-blocks goes up to m.

• ComplAdmTransversals(e,w,core): Returns a list of completely admissible transversals of the e-core of weight w and core core.

• Test1CompltAdmTransversals(e,m): tests if there are (and how many) completely admissible transversals for e-cores up to rank m.

• Test1Proposition7Weight2(k,m): tests how many completely admissible transversals for every e-core with e every odd up to the odd k and every e-core with rank up to m.

• CompTransversals(e,w,core): Returns a list of all the possible compatible transversals for the e-block of weight w of the block corresponding to the core core.