6
votes
3answers
286 views

Representations of S_n induced from centralizers of elements

Does anyone have a reference for a good description of representations of $S_{n}$ obtained by inducing up from $C_{S_{n}}(\pi)$, for some element $\pi$ of $S_{n}$? (I'd prefer an efficient ...
4
votes
4answers
356 views

Structure of the adjoint representation of a (finite) group (Hopf algebra) ?

Every group acts on itself by conjugation $h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation. Question 1: what is known about this representation ...
28
votes
2answers
2k views

What to do now that Lusztig's and James' conjectures have been shown to be false?

Lusztig and James provided conjectures for dimensions of simple modules (or decomposition numbers) for algebraic groups and symmetric groups in characteristic $p$. These conjectures have been ...
15
votes
1answer
455 views

Conjectural identities for Young symmetrizers and Young-Jucys-Murphy elements

The following questions I have found in my own notes from about 3 years ago. Unfortunately, I lost much of the context; I believe I made these conjectures reading Okounkov-Vershik, arXiv:0503040v3, ...
7
votes
1answer
384 views

On Applications of Murnaghan Nakayama Rule

This question is crossposted at math.stackexchange here and may be beyond the usual scope of the site. The question is located below. In short, I am looking for an accessible explanation of the ...
3
votes
1answer
190 views

Combinatorics of index sets multiplicities in characters of symmetric groups

Hi everyone. I'm pondering the following question: I have a Coxeter group $(W,S)$ of type $A_{n-1}$, i.e. the symmetric group $W=Sym(n)$ with the neighbour transpositions as generating set $S=\lbrace ...
2
votes
2answers
259 views

The Jantzen-Schaper theorem

Does anybody have an electronic copy of Schaper's PhD thesis: K.D. SCHAPER, ‘Charakterformeln fur Weyl-Moduln und Specht-Moduln in Primcharacteristik’, Diplomarbeit, Bonn, 1981. I would like to ...
4
votes
1answer
191 views

Expression of basis vectors of permutation modules in different bases.

This is a cross-post from math.se, because I did not get any answer there: Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that ...
4
votes
1answer
162 views

Representations of Sym(n) and SL_d

Irreducible representations of the symmetric group Sym$(n)$, and degree-$n$ algebraic representations of SL$_d(\mathbb C)$ for $d\ge n$, can both be classified by Young diagrams with $n$ boxes. ...
7
votes
1answer
350 views

Permutation character of the symmetric group on subsets of certain size

The symmetric group $S_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simply by $$(g,A)\mapsto ...
1
vote
0answers
89 views

Next smallest dimension of Specht Module after $(n)$, $(1^n)$, $(n-1,1)$ and $(2,1^{n-2})$

In representation theory of $S_n$, we know that for $n \geq 9$, the only Specht modules $S^\alpha$ of dimension $f^\alpha < {n-1 \choose 2} - 1$ are: $S^{(n)}$ and $S^{(1^n)}$ with dimension $1$, ...
5
votes
1answer
194 views

Dimension of Specht Modules $S^\lambda$

In the study of representation theory of $S_n$, we know that the irreducible characters of $\chi_\lambda$ of $S_n$ are indexed by partitions $\lambda \vdash n$. There are several methods in ...
6
votes
1answer
192 views

Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module

Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...
3
votes
2answers
301 views

A basis for Schur functors

Suppose $V$ is a finite-dimensional vector space (over $\mathbb{C}$) and $\lambda$ is a partition of $n$ (not necessarily the dimension). Let $S^\lambda(V)=(V^{\otimes n})_\lambda$ be the $\lambda$'th ...
13
votes
4answers
594 views

Largest permutation group without 2-cycles or 3-cycles

The largest permutation group without 2-cycles is $A_n$, which has size $n!/2$. I think the largest permutation group without 2-cycles or 3-cycles is much smaller, but I can't figure out if it should ...
7
votes
1answer
368 views

Littlewood Richardson rule and seminormal basis of Specht modules

Background Seminormal Basis of Specht modules of $\mathfrak{S}_n$ Let $\lambda$ be a partition of $n$. A $\lambda$-tableau is a bijection $\mathfrak{t}:\lambda \to \{1,2,...,n\}$. We say a ...
4
votes
1answer
223 views

Identity involving partitions coming from representations of alternating groups

It is not difficult to show that the number of conjugacy classes in the alternating group $A_n$ is given by classes in the alternating group = no. of even partitions + no. of self-transpose ...
6
votes
1answer
258 views

Dual of a Specht module

For a partition $\mu$ of $n$, let $S^{\mu}$ be the associated Specht module, defined over $\mathbb{Z}$. For any field $k$, we can tensor $S^{\mu}$ with $k$ to get a representation $S^{\mu}_k$ of the ...
5
votes
2answers
384 views

A product identity for partitions

For a partition $\lambda=(\lambda_1\ge \lambda_2\ge \dots)$, let $m_\lambda=\prod_i (\lambda_i-\lambda_{i+1})!$ be the product of factorials of consecutive differences and let $v_\lambda=\prod_{i | ...
1
vote
1answer
367 views

Schur Weyl duality for sl_n representations

Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl ...
2
votes
0answers
119 views

Partitions limit shape and LDP

Hello! I am trying to understand the paper of Dembo-Vershik-Zeitouni, Large deviations for integer partitions. I am only interested in Theorem 2, which deals with the case of the uniform ...
6
votes
0answers
167 views

Finding a maximal tableau for a sum of Jucys-Murphy Elements

Let $X \in \mathbb{C}[S_n]$ be an element of the group algebra of $S_n$ expressible as the sum of some Jucys-Murphy elements. Then let $\lambda$ be any irreducible representation of $S_n$, with the ...
5
votes
0answers
333 views

What are the relation between Rep(G) and Rep(S_n)?

Let G be a finite group. We know it can be written as a subgroup of S_n. On the other hand, people sometimes say Rep(G) --- the category of all finite dimensional representations, are more interesting ...
1
vote
1answer
200 views

$\lambda$-rings and hopf-rings

The direct sum of complex representation rings $R_*\oplus R\Sigma_n$, for $\Sigma_n$ the $n$th symmetric group is also the free $\lambda$-ring on one generator. Here, we take a product obtained from ...
17
votes
2answers
617 views

An n!-dimensional representation of the symmetric group S_{n+2}

I have come across a sequence of representations $V_n$ of the symmetric group $S_{n+2}$ which has the property that restricting the action $S_n \subset S_{n+2}$ gives the regular representation: $$ ...
18
votes
5answers
1k views

Why are Jucys-Murphy elements' eigenvalues whole numbers?

The Jucys-Murphy elements of the group algebra of a finite symmetric group (here's the definition in Wikipedia) are known to correspond to operators diagonal in the Young basis of an irreducible ...
1
vote
1answer
174 views

Formula for the “integral form” action of Iwahori-Hecke algebra on the standard basis for Specht modules

Is there a formula somewhere in the literature for the action of the generators $T_1,\ldots,T_{n-1}$ of the Iwahori-Hecke algebra on the standard basis of its Specht modules? It is well-known that the ...
5
votes
1answer
462 views

Tensor products of permutation representations of symmetric groups.

I am looking for a reference for the following fact which must be classical (which makes it harder, for me, to track a reference down). I am interested because there are similar (more complicated) ...
3
votes
2answers
500 views

What is known about the centraliser of the Hecke algebra in the affine Hecke algebra?

This question is a sequel to 66602 The Hecke algebra is the quotient of the group algebra of the braid group of type $A_n$ by quadratic relations and the affine Hecke algebra is the quotient of the ...
6
votes
1answer
455 views

Interpreting the Garnir relations in terms of the Yang-Baxter equation

One possible construction of the Specht modules goes as follows. Given a partition $\lambda$ of $n$, we can write down Young's seminormal form for the representation of $S_n$ corresponding to ...
2
votes
0answers
203 views

Picking $n$ so that certain Schur functors of the standard representation of $S_n$ are linearly independent

Let $V_n$ be the standard permutation representation of the symmetric group $S_n$, and let $\mathbb{S}_{\lambda}$ denote the Schur functor associated to the partition $\lambda$. Let $\lambda$ range ...
18
votes
4answers
908 views

What are the applications of immanants?

Definitions of determinant: $det(A) = \sum_{\sigma \in S_n}(-1)^{\operatorname{sgn} \sigma}\prod_{i}a_{i, \sigma(i)}$ and permanent: $perm(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, \sigma(i)}$ ...
6
votes
1answer
715 views

Irreducible decomposition of tensor product of irreducible $S_n$ representations

Are there well known results on the irreducibles in the decomposition of tensor products of irreducible $S_n$ representations? I would also like to know of some references where I can find formulas ...
4
votes
0answers
171 views

Frobenius norms of Fourier coefficients of the symmetric group

Suppose $f \colon S_n \rightarrow \mathbb{R}$ is some weighted collection of permutations. We want to understand how "well-spread" $f$ is. Our first test is its actions on singletons - are the ...
6
votes
1answer
1k views

What is a Specht module?

I'm studying the structure of the Specht module for $S_n$ and I would like to know if there is some generalizations of this structure for Weyls groups or Coxeter groups. Also, I'm interest to know ...
5
votes
1answer
568 views

Decomposition of induced representations in S_n

Let C be a cyclic subgroup of S_n. How does the representation $Ind_C^{S_n}\rho$, where $\rho$ is some representation of $C$, decompose into irreducible components? Is there are a way to know which ...
4
votes
2answers
864 views

Double coset representatives and structure of hecke algebras

Let $GL_n(F_q)$ be the general linear group over finite field $F_q$ and $B_n$ be its borel subgroup consisting of all upper triangular matrices. Then the double cosets $B_n\backslash GL_n(F_q)/B_n$ ...
11
votes
2answers
586 views

Branching rule from symmetric group $S_{2n}$ to hyperoctahedral group $H_n$

Embed the hyperoctahedral group $H_n$ into the symmetric group $S_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S_{2n}$ by ...
3
votes
1answer
230 views

Decompositions for Symmetric Groups

I'm looking for information about how representations of $S_n$ decompose under restriction. I know about the branching rule: That is, in characteristic 0, irreducible modules $L(\lambda)$ for $S_n$ ...
4
votes
2answers
775 views

A sum involving irreducible characters of the symmetric group

Recently, during the research, I came across a sum, denoted by $H(n,L)$, involving irreducible characters of the symmetric group, \begin{equation} H(n,L)\colon=\sum_{Y_{i,j,w}} ...
12
votes
0answers
691 views

Diagonalizing some matrices arising from Fourier transform on $S_n$.

Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...
4
votes
1answer
596 views

some confusion about the explicit construction of irreducible representations of $S_n$

In this book chapter, the irreducible representations of the symmetric group $S_n$ is given in terms of polytabloids of a Ferrer's diagram $\lambda$, defined as $e_t = \sum_{\pi \in C_t} ...
10
votes
4answers
1k views

Making the branching rule for the symmetric group concrete

This question concerns the characteristic $0$ representation theory of the symmetric group $S_n$. I'm a topologist, not a representation theorist, so I apologize if I state it in an odd way. First, ...
8
votes
2answers
963 views

How ugly is the isomorphism R[GxH] = R[G] (X) R[H] for groups G, H?

Clearly, when $G$ and $H$ are two finite groups, and $V$ and $W$ are two representations of $G$ and $H$, respectively, then $V\otimes W$ is a representation of the group $G\times H$. It is a ...
6
votes
1answer
774 views

schur Weyl duality in positive characteristic

Let $S_k$ be the symmetric group. Let $F$ be an algebraically closed field. Let $Rep(S_k)$ be the category of representations of $S_k$ over $F$. Let $Rep(GL_n(F))$ be the category of algebraic ...
3
votes
1answer
348 views

What is a concomitant (and other questions on D.E. Littlewood's “Products and plethysms of characters with orthogonal, symplectic, and symmetric groups” )?

I'm trying to understand the paper "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" by D.E. Littlewood (link), but I'm having trouble overcoming the language ...
11
votes
0answers
398 views

Where do stable Kronecker coefficients live “in nature”?

Background: For a partition $\lambda$, let $\lambda[N] = (N - |\lambda|, \lambda_1, \lambda_2, \lambda_3, \dots)$, also let $\chi_\lambda$ be the corresponding irreducible character of the symmetric ...
7
votes
1answer
490 views

Explicit computation of induced modules of semidirect products with the symmetric group

I've gotten stuck in a project I have been working on, essentially on the following combinatorial question about the symmetric group. One can obtain a 1-dimensional representation $M^n_c$ of the ...