All Questions
71 questions
3
votes
0
answers
73
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
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
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
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 ...
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
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
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
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
0
votes
1
answer
349
views
Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)
Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?
There are two sets of partition polynomials, not in the OEIS, that serve as the ...
- 10.5k
7
votes
1
answer
185
views
Origin of the abacus bijection
What is the origin of the abacus bijection (aka the rim hook bijection, aka the Stanton-White bijection, aka James's bijection)?
Igor Pak, in his 2000 article "Ribbon tile invariants" (...
- 19.7k
6
votes
0
answers
365
views
Is this just a numerical accident or what?
In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation
$$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m}
=\prod_{...
- 43.2k
4
votes
0
answers
163
views
An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
- 43.2k
2
votes
0
answers
87
views
Reference request on Plancherel measure for partitions whose parts differing by more than $1$
Given an (unrestricted) integer partition $\lambda$ of $n$, let $f_{\lambda}$ denote the number of standard Young tableaux (SYT) of shape the Young diagram $Y(\lambda)$ of $\lambda$. Then,
$$\sum_{\...
- 43.2k
5
votes
2
answers
1k
views
What is the motivation behind symplectic/orthogonal content?
Here $\lambda'$ is the conjugate partition of $\lambda=(\lambda_1,\lambda_2,\dots)$ and cells are in the Young diagram.
The symplectic content of cell $(i,j)$ of $\lambda$ is defined by
$$c_{sp}(i,j)=\...
- 43.2k
11
votes
2
answers
977
views
Reference for combinatorics with view towards representation theory/algebraic geometry
I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
- 1,061
4
votes
1
answer
197
views
Moment integrals and determinants
Let $USp(2n)$ be the compact symplectic group of size $2n$, $dA$ its Haar measure
of total mass one, and $\det(1−A)$ being computed for the standard representation of
$A\in USp(2n)$ as a matrix of ...
- 43.2k
0
votes
0
answers
171
views
Total sum of characters over partitions with distinct parts
In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...
- 43.2k
7
votes
2
answers
713
views
Total sum of squares of characters of the symmetric group $\mathfrak{S}_n$
In my earlier MO post, I proposed the double sum $\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}$ regarding characters of the symmetric group $\mathfrak{S}_n$. Soon after, I started ...
- 43.2k
4
votes
1
answer
700
views
Total sum of characters of the symmetric group $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.2k
6
votes
2
answers
613
views
Counting $\pm 1$ and $0$'s in the character tables of $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.2k
2
votes
1
answer
75
views
Reference for the action of the Mullineux involution on a partition with an added good node
Could you help me to locate a reference in the literature to the following fact: if you act by the Mullineux involution $M$ on a partition $Q$ which is a union of a regular partition $P$ with an added ...
2
votes
1
answer
212
views
Changing $S_2 \wr S_n$ for $S_n \wr S_2$ in the theory of zonal polynomials
The permutation group $S_{2n}$ has $H_{2,n}=S_2\wr S_n$ as a subgroup. The plethysm $h_n(h_2)=\sum_{\lambda\vdash n}s_{2\lambda}$ is well known.
The zonal spherical functions $\omega_\lambda(g)=\frac{...
- 1,549
1
vote
0
answers
90
views
Dimension of a certain space of symmetric functions: Part II
This is the second installment of my earlier MO question.
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. Denote the set of all partitions with distinct parts by $\...
- 43.2k
4
votes
0
answers
205
views
Dimension of a certain space of symmetric functions: Part I
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$.
QUESTION. Consider the ...
- 43.2k
2
votes
0
answers
84
views
symmetry for a pair of statistics on partitions
Let $\lambda\vdash n$ denote a partition $\lambda$ of $n$ and let $\square\in\lambda$ denote a box $\square$ in the Young diagram of $\lambda$.
QUESTION. Can you list a pair of (distinct) statistics $...
- 43.2k
10
votes
1
answer
299
views
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.
Let $N^+$ denote the space of uni-upper-triangular ...
- 24.2k
14
votes
2
answers
897
views
A canonical bijection from linear independent vectors to parking functions
Call an $n$-vector $v$ in $\mathbb{Z}^n$ cool when it has only entries 0 or 1 and the ones appear in only one block. Thus there are $n(n+1)/2$ such vectors. For $n=3$ they are:
[ <[ 1, 0, 0 ]>, &...
- 26.5k
6
votes
1
answer
657
views
Does the purported proof of Rota's conjecture provide an algorithm for calculating the forbidden minors of matroids over arbitrary finite fields?
About six years ago there was a proof announced and later outlined in a notice from AMS. However right now I can only seem to find forbidden minor characterizations for matroids linearly ...
- 2,506
4
votes
1
answer
499
views
I want to know the name of or any references for a matrix in the book "The representation theory of the symmetric groups" by Gordon James
$\DeclareMathOperator{\Ind}{\operatorname{Ind}}$I'm reading "The representation theory of the symmetric groups" written by Gordon James.
I found the matrix $B$ in the chapter 6 ("The ...
- 1,013
10
votes
1
answer
396
views
Generalising the union-closed sets conjecture from lattice to a larger class of posets
(edit: I decided to simplify the question and only pose it for bounded posets first)
The Union-closed sets conjecture is equivalent for lattices P to:
There exists a join-irreducible element $a$ with ...
- 26.5k
4
votes
1
answer
217
views
Computation of the Lusztig a-function
See for example https://www.sciencedirect.com/science/article/pii/0021869387901542 for the definition of the Lusztig a-function.
Question 1: Is there a table for the values of Lusztig's a-function ...
- 26.5k
10
votes
3
answers
1k
views
Number of permutations with longest increasing subsequences of length at most $n$
Is there a known expression for, or a nontrivial upper bound on, the number of permutations in $S_k$ with longest increasing subsequence of length at most $n$?
Let $l(\sigma)$ denote the length of the ...
- 201
3
votes
1
answer
212
views
Seeking a combinatorial proof for the invariance of a $q$-series
Start with some notations: $(a,q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$, shortened by $(a)_n$, and $(a)_{\infty}=\prod_{k=0}^{\infty}(1-aq^k)$.
It's easy to verify (using algebraic means) that, for each $...
- 43.2k
8
votes
1
answer
512
views
RSK correspondence
Up to now, what are the difference ways we know to define RSK correspondence? I already know:
By insertion and recording tableau.
Ball construction or Viennot's geometric construction.
Growth diagram ...
- 320
4
votes
0
answers
123
views
Bijections of Littlewood-Richardson coefficients
Let $c^{\lambda}_{\mu\nu}$ be the Littlewood-Richardson coefficients, where $\lambda,\mu,\nu$ are partitions. We know that $c^{\lambda}_{\mu\nu}= c^{\lambda}_{\nu\mu}$. Up to now, what are the ...
- 320
3
votes
0
answers
123
views
Tableaux switching
I'm reading the article Tableau Switching: Algorithms and Applications by Benkart, Sottile, and Stroomer. Do you know if there are any articles or books that talk more about the properties of tableau ...
- 320
3
votes
0
answers
87
views
References for Littelmann Path models and related combinatorial objects
I am looking for a reference (preferably textbook or lecture notes) having Littelmann path models and it's uses in representation theory, combinatorics. I am aware of the original papers by Littelmann....
- 427
3
votes
0
answers
101
views
Hermitian sublattices of a given type
Consider an unramified quadratic extension $E/F$ of non-archimedean local fields, and suppose that $\langle\cdot,\cdot\rangle$ is a fixed Hermitian form on $E^d$ such that $\mathcal{O}_E^d$ is self-...
- 2,516
1
vote
1
answer
154
views
$q$-plane partitions & specialization & interlinks
MacMahon's enumeration of all plane partions (PP) inside an $n$-cube generalizes to
$${\tt PP_n}(q)=\prod_{i,j,k=1}^n\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}.$$
A $q$-analogue of symmetric plane partitions ...
- 43.2k
10
votes
3
answers
828
views
The vanishing of sum of coefficients: symmetric polynomials
Denote $\pmb{X}_n=(x_1,x_2,\dots,x_n)$. Consider the symmetric polynomial
$$f_n(\pmb X_n)=\prod_{1\leq i<j\leq n}(x_i+x_j).$$
Expand these in terms of elementary symmetric polynomials, say
$$f_n(\...
- 43.2k
7
votes
1
answer
290
views
Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials
$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}_n$ be the symmetric group, $\SYT_n$ be the set of standard young tableaux of size $n$.
For $u\in \...
30
votes
1
answer
2k
views
Is there an accessible exposition of Gelfand-Tsetlin theory?
I'm hoping to start an undergraduate on a project that involves understanding a bit of Gelfand-Tsetlin theory, and have been tearing my hair out looking for a good reference for them to look at. ...
- 44.7k
2
votes
0
answers
271
views
About relation between Kostka numbers and Littlewood-Richardson coefficient
The fact that Kostka numbers equals to Littlewood-Richardson coefficients for some partitions is already known $\colon$
\begin{align}
K_{\lambda \mu} = c_{\sigma \lambda}^\tau
\end{align}
where $\...
- 31
17
votes
2
answers
742
views
Equivariant Möbius inversion
I'll first explain what Möbius inversion says, and then state what I am fairly sure the equivariant version is. I can write out a proof, but I also can't believe this hasn't been done already; this is ...
- 156k
8
votes
1
answer
203
views
Reference request: Coxeter length and irreducible characters
Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have
$$\ell(\pi) = |\...
- 809
6
votes
2
answers
429
views
Natural bijection between Dyck paths and tilting modules
Let $A_n=kQ_n$ be the path algebra of linear oriented Dynkin graph $Q_n$ of Dynkin type $\mathcal{A_n}$ (so $A_n$ is the unique hereditary Nakayama algebra given by quiver and relations).
The number ...
- 26.5k
4
votes
0
answers
76
views
Comparing parametrizations of unipotent radical
Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. ...
- 2,516
3
votes
1
answer
225
views
$f^{\lambda}$: asymptotics and analytic continuations
Let $\mathbb{Y}_n$ denote the set of all partitions of $n\in\mathbb{N}$ and $\mathbb{Y}$ Young's lattice of all partitions. The partition function $g_0(n)=\sum_{\lambda\in\mathbb{Y}_n}1$ has an ...
- 43.2k
2
votes
1
answer
431
views
Lagrange interpolation vs homogeneous symmetric polynomials?
This question is a follow-up on another MO query here.
Question. For $r\geq$ an integer, is it true that there exists homogeneous symmetric polynomial $P_r(x_1,\dots,x_n)$ with positive ...
- 43.2k