3
votes
0answers
67 views

Orthogonal basis for the multilinear polynomials with zero “trace”

We say that a multilinear polynomial $P(x_1,\ldots,x_n)$ in $n$ commuting variables over $\mathbb{R}$ has zero trace if $$ \frac{d}{dt} P(t,\ldots,t) = 0. $$ Equivalently, $$ \left(\sum_{i=1}^n ...
3
votes
0answers
119 views

characterization of all periodic tiling of a simple set of Wang Tile

Consider a set of Wang Tile such that all the edges are either 1 or 0.... there are 16 elements in such a set. Now, I wish to characterize all the periodic tilings of this set (better if they are ...
2
votes
1answer
189 views

Question about a proof in Graham and Lehrer's “Cellular algebras”

I'm sorry if this question is too basic for MO. I'm reading a paper by Graham and Lehrer "Cellular algebras" and have trouble understanding one step in a proof of a crucial theorem. I suppose that the ...
14
votes
0answers
349 views

Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...
0
votes
0answers
141 views

special values of L-functions cohen-lenstra heuristic

I found some lecture notes on links between number theory and random permutations. It was difficult to follow: The notes start with an interesting fact, whose proof I've asked on Math.StackExchange: ...
9
votes
0answers
174 views

Degree of a cone over the set of rank $r$ $n\times n$ matrices

Let $X_r\subset Mat_{n\times n}(C)$ denote the variety of rank at most $r$ matrices, set $k=n-r$ and assume $n\geq k^2-1$. Consider the cone over $X_r$ with vertex spanned by the first $k^2-1$ entries ...
11
votes
1answer
496 views

Counting representations of $k[x,y]$ when $k$ is finite

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial ...
12
votes
2answers
406 views

calculating Littlewood-Richardson coefficients

It is known that if $\alpha,\beta,\gamma$ are three partitions then the Littlewood-Richardson coefficient $c_{\alpha \beta}^{\gamma}$ is positive when the triple ($\alpha,\beta,\gamma$ ) occurs as ...
7
votes
1answer
256 views

Iterated Pieri's rule, Schur functors and intersection of subrepresentations

Let $\lambda$ and $\mu$ be two Young diagrams, such that $\lambda$ can be obtained from $\mu$ by extending one single column with additional $b$ boxes. Let $\Sigma^\lambda U$ and $\Sigma^\mu U$ denote ...
5
votes
0answers
150 views

Is there any natural construction for the irreducible representations of $G\wr S_n$?

For $S_n,$ one can construct all the irreducible representations through the young diagrams. Is there any natural construction for the irreducible representation of $G\wr S_n$ (G is a finite group)?
7
votes
1answer
134 views

Help identify this generalized sign of real representations

Let $V$ be a real representation of a finite group $G$. Define $\mathbb Z[I]_{I\leq G}$ to be the ring over the integers generated by subgroups of $G$ with multiplication corresponding to ...
15
votes
1answer
517 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, ...
5
votes
0answers
134 views

Is there an Arctic Circle phenomenon for Amman-Beenker tilings?

I found some slides on tilings and one of them pertained to Amman-Beenker tilings. It looks like there is an Arctic Circle phenomenon similar to that for dominos or lozenges. Is there any ...
7
votes
1answer
464 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 ...
4
votes
0answers
85 views

Differences of Numbers of Helicity States in 4-dimensional Strings

The question whether the states in $D=2m + 2$ dimensional string theory, which carry a representation of $SO(2m)$, span spaces which carry representations of $SO(2m+1)$ seems hopelessly complicated. ...
0
votes
1answer
91 views

Previous derivation of a combinatorial formula for the fundamental representations of $A_{n-1}$

Consider the simple Lie algebra $A_{n-1}$ over the complex numbers. I refer to $A_{n-1}$ rather than $A_n$ for reasons that I try to make clearer below. The following facts are no doubt well known, ...
0
votes
0answers
110 views

Semi-Standard Young Tableaux: Do Diagrams for $O(2m)$ combine to Diagrams from $O(2m+1)$?

Let $n_\lambda^K$ be the number all semi-standard Young tableaux of size $K$ with Ferrers diagrams diagram $\lambda$ (i.e. the number of all fillings of $\lambda$ with natural numbers with weakly ...
4
votes
2answers
289 views

Symmetric powers of Schur polynomials

I apologize if this question is trivial, but a couple of days of searching for necessary routines have led me here. Does there exist software to compute symmetric powers of Schur polynomials? ...
10
votes
2answers
283 views

Temperley-Lieb algebras for other Weyl groups?

The Temperley-Lieb algebra has the same generators as the $S_n$ group algebra, and the same commuting relations, but its other relations are different. A nice diagrammatic interpretation can be seen ...
3
votes
1answer
193 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 ...
8
votes
0answers
406 views

Reference/quote request: “All of combinatorics is the representation theory of $S_n$”

I think I remember reading somewhere a glib (or is it deep?) quote, perhaps due to Rota?, which was something like "All of combinatorics is essentially [or can be reduced to?] the representation ...
2
votes
0answers
131 views

Summing Characters of the Symmetric Group over Derangements (Enumerative Combinatorics: Vol. II Ex. 7.63)

The following exercise is out of Stanley's Enumerative Combinatorics: Vol. II (Ex. 7.63): For $\lambda \vdash n$ define $d_\lambda = \sum_{w \in D_n} \chi^\lambda(w)$ where $D_n$ is the set of all ...
8
votes
3answers
531 views

Is there a list of Kazhdan-lusztig polynomials?

When studying the combinatorics and representation of Coxeter groups, I often find it irksome to compute KL- and R- polynomials from scratch on Maple or Sage. The time it consumes to generate a ...
4
votes
1answer
399 views

Hilbert Matrix and Approximation Theory

I was reading about the Hilbert matrix and Cauchy determinants: \[ \det \left[ \frac{1}{i+j-1} \right]_{i,j} \] By guessing where this determinant is $0$ or $\infty$ we can guess the right formula. ...
1
vote
0answers
91 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 $1$, ...
4
votes
0answers
151 views

Eigenvalues of “modified” Johnson scheme via the representation theory of the symmetric group

I am interested in eigenvalues of the following association scheme, which somewhat resembles the Johnson scheme. Let $n$ and $k\leq n$ be positive integers. The $n!/(n-k)!$ vertices of the scheme ...
2
votes
1answer
172 views

A question on Lusztig's `graph with automorphism' construction?

Using the notion of a graph with compatible automorphism, Lusztig constructs all symmetrizable Cartan data (i.e. Cartan matrices $A$ for which there is a diagonal matrix ...
2
votes
1answer
177 views

PROPs representations, free module analog

Ordinary operad with one ouput can be obviously regarded as free module on itself. Is there are analogous construction for operad with many outputs (PROP)? This must be difficult question, but what is ...
4
votes
1answer
322 views

Known decomposition of $\bigwedge^k Sym^d \mathbb C^n$ in special cases?

Let $V = \mathbb C^n$. Consider the plethysm $\bigwedge^k Sym^d V$ as a representation of $GL(V)$. In what special cases (e.g., for what $k$, $d$, and $n$) is this representation's decomposition into ...
2
votes
0answers
130 views

Harmonic analysis and non-symmetric Macdonald polynomials?

I have recently been reading a lot about Macdonald polynomials, the symmetric and the non-symmetric ones. One thing that strikes me is that the symmetric Macdonald polynomials admit a positive theory, ...
13
votes
4answers
611 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 ...
17
votes
1answer
1k views

Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order ...
7
votes
1answer
394 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 ...
1
vote
2answers
505 views

Is there formula name and proof for this theorem ? [closed]

The formula answers: how many tuples $(\sigma_1,\sigma_2,...,\sigma_n)$ of elements of a given group G such that (1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class. (2) ...
3
votes
0answers
245 views

det(A)det(B) = det(AB+correction), Capelli identities, “factorzied” representation of gl_n

Context: some probably know that there are Capelli identities which state det(A)det(B) = det(AB+correction) for some matrices with non-commuting elements, they go back to 19-th century, but also ...
15
votes
0answers
401 views

Cauchy matrices with elementary symmetric polynomials

$\newcommand{\vx}{\mathbf{x}}$ Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by \begin{equation*} e_k(\vx) := \sum_{1 ...
34
votes
1answer
859 views

Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?

In their seminal 1979 paper here, Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corresponding Iwahori-Hecke algebra. In particular they showed how to pass from a standard ...
2
votes
2answers
418 views

elements in the weyl group

Let W be the Weyl a group of a semisimple simply connected group over C. Let I={1,...,r} the set of simple roots. For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
4
votes
1answer
228 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 ...
8
votes
1answer
290 views

On q-Demazure operators

Setup Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic with Borel subgroup $B$. Let $\Lambda$ denote the weight lattice of $G$; we write elements ...
11
votes
3answers
1k views

Tensor powers of the standard representation

Consider $V_{(n-1, 1)}$, the $n-1$ dimensional irreducible representation of $S_n$, i.e. the "standard" or "defining" representation. Is there a nice formula for how the $k$-th tensor power of ...
8
votes
0answers
257 views

Chern Classes of Exterior Products of a vector bundle.

This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of ...
7
votes
1answer
333 views

Explicit method to compute Macdonald/Koornwinder functions

I'd like to compute explicitly symmetric Macdonald functions associated to arbitrary (possibly non-reduced) root systems, using Computer Algebra System. Unfortunately Sage seems to only implement ...
11
votes
1answer
504 views

Reconstruction Conjecture holds for Directed Acyclic Graphs?

Wikipedia's article on the Reconstruction Conjecture mentions that the conjecture is false for digraphs, and refers to two papers by Stockmeyer. As far as I can see, none of the counter-examples in ...
5
votes
2answers
401 views

Can one recognize this symmetric function?

$\newcommand{\lm}{\lambda}$ $\newcommand{\bR}{\mathbb{R}}$ Let $m$ be an integer $>1$. Define $$ I_m:\bR^m\to \bR,\;\; I_m(\lm_1,\dotsc, \lm_m)=\int_{S^{m-1}}\exp\Bigl(-\sum_{j=1}^m ...
13
votes
1answer
396 views

Categorifying the equality of product and coproduct of symmetric functions

Littlewood-Richardson coefficients are both multiplicities of $GL_n$ tensor products, and of restrictions of $GL_{m+n}$ representations to $GL_m \times GL_n$. I want to turn this equality of numbers ...
5
votes
2answers
391 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 | ...
3
votes
2answers
1k views

numbering the squares of a rectangular grid, was: counting sequences of pairs

Hi, Barry Cipra has rephrased my question in far superior clarity and brevity in an addendum to his answer below. I quote: "If you number the squares of an $m×n$ grid, you can let three groups act ...
10
votes
1answer
372 views

Schur functors generalization to “Jack”, “Hall-Littlewood”, “Macdonald” functors ?

Schur functors are functors from the category of vector spaces to itself. If we take an operator $M: V->V$ and apply a Schur functor to it and then calculate trace $Tr(M^{\Lambda})$ we will get ...
7
votes
2answers
459 views

What is the tropical Robinson-Schensted-Knuth correspondence?

And what are it's applications? A conceptual explanation would be great! Is there an expository note about this somewhere? Some references have already appeared in the answers and comments below. To ...