**1**

vote

**1**answer

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 ...

**8**

votes

**1**answer

245 views

### Restriction of characters of hyperoctahedral groups.

The hyperoctahedral group $H_n$ has several descriptions; as a wreath product; as signed permutation matrices; as the Weyl group of type $B_n$ or $C_n$. In all these descriptions it is apparent that ...

**14**

votes

**3**answers

1k views

### Rhombus tilings with more than three directions

The point of this question is to construct a list of references on the following subject: Fix vectors $v_1$, $v_2$, ..., $v_g$ in $\mathbb{R}^2$, all lying in a half plane in that cyclic order. I am ...

**67**

votes

**3**answers

2k views

### What do the stable homotopy groups of spheres say about the combinatorics of finite sets?

The Barratt-Priddy-Quillen(-Segal) theorem says that the following spaces are homotopy equivalent in an (essentially) canonical way:
$\Omega^\infty S^\infty:=\varinjlim~ \Omega^nS^n$
...

**7**

votes

**1**answer

301 views

### Two ways of generalizing factorials via symmetric groups

By the Bruhat decomposition of $GL(n, \mathbb{F}_q) / B_n$ we know that $$[n]! = \sum_{ \sigma \in S(n)} q^{l(\sigma)}$$ where $[n]! = \prod_{j=1}^n (1+q + \cdots + q^{j-1})$ and $l(\sigma)$ is the ...

**20**

votes

**1**answer

583 views

### Is there an easy description of the structure of this infinite group?

Let $S_\infty$ denote the full symmetric group on countably many points and index the points by $\mathbb{N}$. For any weight (for lack of a better name) function $w:\mathbb{N}\rightarrow\mathbb R^+$, ...

**5**

votes

**1**answer

463 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) ...

**1**

vote

**2**answers

374 views

### Difference between orthogonal form and seminormal form

Frequently in the literature on Hecke algebras for the symmetric group and their generalisations, one encounters references to Young's seminormal form and Young's orthogonal form. I have a good ...

**3**

votes

**2**answers

501 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

**1**answer

456 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

**0**answers

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 ...

**19**

votes

**4**answers

913 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

**1**answer

716 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 ...

**0**

votes

**2**answers

486 views

### Lie algabra of symmetric group

It's easy to see that the descending central series of a group induces a graded Lie algebra .(see for example Serre's Harvard lectures or Magnus-Solitar book). I think in general this can be ...

**9**

votes

**4**answers

1k views

### Cyclic Permutations - but not what you think

This question is not about elements of $S_n$ that consist of a single $n$-cycle, though naturally it's related.
Instead, consider permutations modulo the action of $(123\ldots n)$. That is, we ...

**4**

votes

**0**answers

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 ...

**2**

votes

**2**answers

430 views

### Convolution on symmetric group Sn

I have question regarding convolution of functions (say
g and h) defined on Sn. In Fourier space this is equivalent to IFT(G.H),
where G = FT(g) and H = FT(h).
Fast Fourier transforms (Clausen's FFT) ...

**5**

votes

**1**answer

527 views

### sum of the character of the symmetric group

Suppose $\mu$ is a fixed partition of $n$ of length $l(\mu)$, and I was encountered with the following sum, namely
$\sum_{\nu} \chi_{\nu}(\mu)$.
I did some calculation using the character table that ...

**6**

votes

**1**answer

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

**1**answer

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 ...

**1**

vote

**1**answer

417 views

### Reference for Hecke algebra version of Young's orthogonal basis

In the paper Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Arun Ram defines a seminormal basis as follows:
given a chain of split semisimple $K$-algebras $K\cong H_0 \subseteq ...

**4**

votes

**2**answers

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 $B_n\backslash GL_n(F_q)/B_n$ ...

**6**

votes

**4**answers

574 views

### Can this nested sum be expressed in terms of generalized harmonic numbers and the cycle index polynomials of the symmetric groups?

For a paper I was working on recently I needed to find the value of the following sum:
$$S(n,k) = \sum_{i_1 = 1}^n \sum_{i_2 = i_1+1}^n \cdots \sum_{i_k=i_{k-1}+1}^n \frac{1}{i_1 i_2 \cdots i_k}.$$
...

**11**

votes

**2**answers

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 ...

**13**

votes

**2**answers

1k views

### roots of permutations

Consider equation $x^2=x_0$ in symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 1$, the maximal number of solutions (square roots) has identity permutation? ...

**12**

votes

**5**answers

1k views

### A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$

$\def\S{\mathbb{S}}$ Dear all,
So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is:
for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$
Is there a general ...

**3**

votes

**1**answer

231 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

**2**answers

777 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

**0**answers

692 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

**1**answer

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

**4**answers

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

**2**answers

964 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 ...

**2**

votes

**0**answers

246 views

### Function recursion relation over symmetric group

Hi!
Let P be a permutation in the symmetric group SN and let π=πj, j+1 be a transposition of elements j and j+1 of the permutation. Let A(P) be a function in dependence of the permutation P. ...

**0**

votes

**1**answer

486 views

### Hankel determinants of symmetric functions

The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions.
...

**6**

votes

**1**answer

783 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

**1**answer

349 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

**0**answers

399 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 ...

**6**

votes

**3**answers

1k views

### Sarrus determinant rule: references, extensions

SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS
An undergraduate came to me with an identity for 4x4 determinants that is actually correct:
$\det(A)=h(A)+h(RA)+h(R^{2}A)$
where R cyclically ...

**2**

votes

**2**answers

137 views

### Algebraic structures at hypernatural parameters

Let's start with some family of algebraic structures of the same type indexed by the natural numbers, say the symmetric group $S_n$. Suppose that the axioms of this algebraic structure (in this case, ...

**10**

votes

**1**answer

336 views

### Reference request: The stable Kronecker ring is isomorphic to the ring of symmetric polynomials

Background
For $\lambda$ any partition and $n$ a positive integer, write $\lambda[n]$ for the sequence $(n - |\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_r)$. For $n$ large enough, this is a ...

**7**

votes

**1**answer

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 ...