All Questions
Tagged with co.combinatorics rt.representation-theory
526 questions
0
votes
1
answer
114
views
Coxeter matrix of Dyck path
I am trying to understand Gjergji Zaimi's answer in What are the periodic Dyck paths?. In the third paragraph he claims that
Next, we define the matrix $X_D$
similarly to the Cartan matrix except we ...
3
votes
0
answers
70
views
While expanding Jack polynomials in monomial basis
Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ ...
- 43.2k
6
votes
0
answers
129
views
Is there a more natural way to define the Young symmetrizer and the Specht module?
It's well known that Young symmetrizer is a fundamental tool in the representation theory of symmetric groups.
For instance, for every Young diagram $\lambda\vdash n$, we construct a Young tableau $T_\...
4
votes
0
answers
81
views
Classification of nilpotent orbits over local fields (for type ABCD via partitions )
Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent ...
- 6,622
0
votes
0
answers
95
views
Class multiplication coefficients of symmetric groups
My question is that I was working with some counting problems, and finally the answer should be
$$
\nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\...
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
3
votes
0
answers
169
views
Basis of Specht module of symmetric groups
I am reading the construction of the Specht module from James's book. The Specht module of a symmetric group corresponding to a partition $\lambda$ is spanned by all polytabloids $e_{t}$ associated ...
- 179
2
votes
0
answers
111
views
Map between Weyl modules in terms of Young tableaux
The irreducible algebraic representations of $\text{GL}_n$ over the complex numbers are given by highest weight representations of dominant weights $\lambda=(k_1,k_2,\ldots,k_n): k_1 \ge k_2 \ge \...
- 461
5
votes
1
answer
344
views
Are parabolic Kazhdan-Lusztig polynomials truncations of the usual Kazhdan-Lusztig polynomials?
Let $(W,S)$ be the affine Weyl group associated to a simple root system.
For $x,y \in W$ we have the usual Kazhdan--Lustig polynomials $h_{y,x} \in \mathbb{Z}[v]$ in Soergel's normalisation, and if ...
- 10.5k
11
votes
1
answer
340
views
Number of odd-dimensional irreducible representations of $S_n$
In this paper the structure of odd-dimensional irreducible representations of the symmetric group is described, but what is the asymptotic behaviour of the number of such representations? (Or, if it ...
- 109k
1
vote
0
answers
55
views
Combinatorial structure of the entanglement spectrum and quantum error correction in finite vector spaces
Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ with dimension $d$. Consider a subspace $S \subset V^{\otimes n}$ representing the code subspace of a quantum error correcting code. We ...
- 11
1
vote
1
answer
102
views
Multiplicities and double and triple tensor products of simple $\frak{g}$-modules
Given a complex simple Lie algebra $\frak{g}$ and a simple module $V_{\lambda}$ for some dominant weight $\lambda$. Consider the tensor product decomposition
$$
V_{\lambda} \otimes V_{\lambda} \simeq ...
- 417
5
votes
0
answers
167
views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,351
0
votes
0
answers
77
views
Generating function for dimensions of the space of polynomials fixed by a single permutation
Consider the space of polynomials with complex coefficients $\mathbb{C}[x_1,x_2,\dots,x_n]$ and let $\sigma$ be a permutation of $\{1,2,\cdots, n\}$ that acts on
this space via $\sigma(x_i)=x_{\sigma(...
- 141
1
vote
0
answers
108
views
Schur Weyl Duality for Maximal Torus
I wanted to know if there's some version of Schur Weyl Duality for the maximal torus $T \subset \operatorname{GL}(V)$? Is there also some combinatorial object which might be useful for the same?
3
votes
0
answers
89
views
Young symmetrizers-like projections to the center of group algebra
Let $A:=\mathbb{C}S_n$ be the symmetric group aglebra.
Let $T$ be a standard Young tableaux of shape $\lambda$. Denote $R(T)$ and $C(T)$ as row and column stabilizers of $T$. For a set $S \subseteq ...
- 31
0
votes
0
answers
117
views
An interesting identity involving skew-Schur functions
Denote $\rho=(-\frac12,-\frac32,-\frac52,\dots)$. I was reading this interesting paper, where in particular, the authors claim that one can get the expression in (2.9)
\begin{align*}
\prod_{k\geq1}(1+...
- 43.2k
2
votes
0
answers
45
views
$K_0$-basis modules with a unique extension related to parking functions
Let $B=B_n$ be the quiver algebra of type $A_n$ (with some orietnation) with $n$ points.
A $B$-module $M$ is a basis of the Grothendieck group $K_0$ if $M$ has $n$ indecomposable direct summands and ...
- 26.5k
17
votes
0
answers
402
views
Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )
Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
- 24.9k
3
votes
0
answers
202
views
What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?
Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
- 24.9k
3
votes
0
answers
102
views
A class of bipartite graphs appearing in higher Auslander--Reiten theory
Let $G = (V,E)$ be a simple undirected bipartite graph with vertices $V$, edges $E$, and a chosen partition $V = X \cup Y$.
Recall that the bipartite complement of $G$ is the graph on the same vertex ...
- 66
3
votes
2
answers
220
views
Factorizations of an $n$-cycle in $S_n$ into a product $xy$ where $|x| = 2, |y| = 3$
Let $S_n$ be the symmetric group on $n$ letters. When (and how) can an $n$-cycle in $S_n$ be factored into a product $xy$, where $x,y$ have orders 2,3 respectively?
More precisely, I'd like to ...
- 7,527
1
vote
0
answers
148
views
Exploring the Intersection of Expander Graphs, Number Theory, Representation Theory and Recent Computer Science Developments [closed]
I have a solid understanding of the basics of expander graphs and their properties and the recent development of High-Dimensional Expanders and their application to Random Walks, along with other ...
7
votes
1
answer
379
views
Coefficients when rewriting the Hook-Content polynomials in terms of binomial polynomials
Let $\lambda$ be a partition, $S_\lambda$ the Schur functor attached to $\lambda$, and let $p_\lambda(t)$ be the polynomial determined by the condition that $\dim S_\lambda(k^n) = p_\lambda(n)$ for ...
- 28.1k
1
vote
0
answers
130
views
Relationship between the symmetric group representation (Specht module) of a Young diagram and the Young diagram obtained by deleting one row
Suppose $\lambda$ is a Young diagram, and $\lambda'$ is obtained by deleting one particular row of $\lambda$. Is there any relationship between the symmetric group representation (Specht module) ...
- 25
14
votes
1
answer
660
views
Is this generalized version of plethysm Schur positive?
Question: Suppose that $f(x_1, x_2, \dots x_n)$ is a polynomial with nonnegative integer coefficients. For each permutation $\sigma\in S_n$, let $f_{\sigma}$ denote $f(x_{\sigma(1)}, \dots, x_{\sigma(...
- 85.6k
0
votes
0
answers
217
views
On characters of the symmetric group: Part 2
This question is related to my earlier MO quest.
For an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\...
- 43.2k
2
votes
0
answers
352
views
On characters of the symmetric group: Part 1
Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\...
- 43.2k
15
votes
1
answer
1k
views
Is this a typo in Macdonald's paper "The Poincaré Series of a Coxeter Group"?
I have a question about the proof of lemma 2.14 in Macdonald's paper The Poincaré Series of a Coxeter Group [1], where he used induction on $l(w)$ to prove that if $|E|=|R(w)|$, then $E=R(w)$. The ...
- 565
4
votes
0
answers
160
views
Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram
I have a "very" indirect proof that the following fact is true for every Young diagram $\lambda \vdash n$ and every $r \in \{0,\dotsc,n\}$:
\begin{equation}
d_\lambda = \sum_{a \in \mathrm{...
- 143
3
votes
0
answers
98
views
Isomorphism and counting for tree quivers
Let $Q$ be a quiver which is a connected tree and let $A=KQ/I$ be a quiver algebra with $I$ an admissible ideal, meaning that $I$ is generated by paths of length $\geq 2$. Let $n$ be the number of ...
- 26.5k
3
votes
0
answers
93
views
Commutant of irrep of $S_n$ (over local field)
Let $k$ be a field of characteristic zero and let $(V, \rho)$ be a finite-dimensional representation over $k$ of the symmetric group $S_n$. I would like to understand the commutant $\operatorname{End}...
- 363
4
votes
0
answers
313
views
What is $\dim D^{\lambda}$ for the symmetric group?
What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...
- 571
3
votes
1
answer
221
views
Asymptotics for number of $p$-regular partitions of $n$
The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...
- 571
2
votes
1
answer
107
views
Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces
For the special linear algebra $\frak{sl}_{m}$ which finite dimensional irreducible representations $V_{\mu}$ have non-trivial zero weight spaces?
For $\frak{sl}_2$ this is clear: $V_{2k\pi}$ for $\pi$...
- 157
1
vote
1
answer
114
views
Combinatorics behind certain induction of characters of the Coxeter group of type $B_n$
Let $W_n$ be a Coxeter group of type $B_n$ with $n\geq 1$. Concretely, it is generated by a set of simple reflexions $S = \{s_1,\ldots ,s_n\}$ which satisfy the relations $s_i^2 = 1, s_is_j=s_js_i$ as ...
- 769
37
votes
1
answer
1k
views
Errata for Fulton's "Young tableaux"
Fulton's Young tableaux is one of the best texts on the subject, one which I
often recommend and cite for reference. Unlike Fulton/Lang and
Fulton/Harris,
it is neither an early-dawn draft nor a ...
2
votes
0
answers
60
views
Klyachko-type inequalities for shifted Schur structure constants
Klyachko (+Knutson-Tao) provide a set of inequalities that are necessary and sufficient for the Littlewood-Richardson coefficient $c^{\lambda}_{\mu\nu}$ to be non-zero.
Is there a similar result for ...
- 15.8k
4
votes
1
answer
269
views
Submodule lattices of preprojective algebras
Let $A$ be a preprojective algebra of Dynkin type.
Question 1: Let $P$ be an indecomposable projective $A$-module. It is known that the submodule lattice of $P$ is finite in Dynkin type $A_n$. Does ...
- 26.5k
19
votes
2
answers
851
views
The discriminant of the Okada algebra
The Okada algebra $\mathfrak{O}_n$ over a field $K$ has generators
$E_1,\dots,E_{n-1}$ and relations $E_i^2=x_iE_i$,
$E_{i+1}E_iE_{i+1}=y_i E_{i+1}$, and $E_iE_j=E_jE_i$ for $|i-j|\geq
2$, where $x_i,...
- 50.8k
0
votes
0
answers
116
views
Multivariate polynomial representations of the infinite dihedral group
The presentation given in Wikipedia for the infinite dihedral group is
$$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$
Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...
- 10.5k
2
votes
0
answers
49
views
Minimal faithful modules for hereditary algebras
A module $M$ is minimal faithful if $M$ is faithful and if we remove any indecomposable summand of $M$, then $M$ is not faithful anymore.
Question: Is there a classification of minimal faithful ...
- 26.5k
7
votes
0
answers
176
views
The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial
I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...
- 131
4
votes
0
answers
272
views
Representations of $\mathrm{sl}(3,\mathbb{C})$ and Catalan-like paths
Background on representations of $\mathrm{sl}(3,\mathbb{C})$
In Chapter 6 of Brian C. Hall's book "Lie Groups, Lie Algebras, and Representations", he constructs the irreducible ...
- 484
5
votes
1
answer
268
views
Enumerating monomials in Schur polynomials
Let $s_{\lambda}(x_1,\dots,x_k)$ be the Schur polynomial associated to the partition $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k>0)$.
Among the many things involved with these ...
- 43.2k
3
votes
0
answers
139
views
Root space inner products and the partial order on roots
For a root system $R$ and a choice of positive roots $R^+$ it is a standard fact (see, e.g., Bourbaki, "Lie Groups and Lie Algebras," Theorem 1 of Section 1.3 of Chapter VI) that
if $(\...
9
votes
1
answer
327
views
The convex hull of Schur polynomial evaluations
Let $r\leq n$ and $d$ be positive integers. A probability vector is a vector of non-negative entries that sum to 1. For each probability vector $\lambda$ of length $n$, let
$$s(\lambda)=(\dim[\pi] \...
- 980
1
vote
0
answers
180
views
Applications of hyperbolic polynomials?
The recently posted MO-Q "Positivity of the coefficients of Taylor series associated to the Riemann hypothesis" (see also this MO-Q) has re-kindled my interest in hyperbolic polynomials--...
- 10.5k
2
votes
0
answers
228
views
Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...
- 43.2k
6
votes
1
answer
588
views
A numerical matrix of power sum polynomials
Let $p_i=x_1^i+x_2^i+\cdots+x_m^i=\sum_{k=1}^mx_k^i$ be the power sum polynomials. Then, the determinant of the $m\times m$ Hankel matrix $M_m=(p_{i+j-2})$, for $1\leq i,j\leq m$, has a neat ...
- 43.2k