Skip to main content
MathOverflow

Questions tagged [symmetric-functions]

Ask Question

Symmetric functions are symmetric polynomials, in finitely many, or countably infinitely many variables. They arise in the representation theory of symmetric groups and in the polynomial representation theory of general linear groups. Bases of the ring of symmetric functions are indexed by integer partitions. Schur functions, elementary symmetric functions, complete symmetric functions, and power sum symmetric functions are the most commonly used bases.

369 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
10 votes
3 answers
464 views

Based on some small calculations in SageMath, I conjectured that the Schur expansion of $h_n[h_k]$ contains $s_{(k,k,...,k)}$ if and only if $k$ is even. For example, this is easily seen when $n=2$. ...
Soumyadip Sarkar's user avatar
  • 419
1 vote
0 answers
140 views

Dear mathoverflow community, I would like to understand the fibers of the mapping from $\mathbb{C}^n$ to itself defined by $\phi\colon (z_1, \ldots , z_n)\mapsto (\Pi_1, \ldots , \Pi_n)$, where $\...
Romain Dujardin's user avatar
  • 111
4 votes
0 answers
234 views

For a triple of partitions, $\mu,\nu, \lambda$, one can consider the Littlewood-Richardson coefficient $c_{\mu,\nu}^{\lambda}$. e.g. $c_{31,21}^{421}=2$. Richard Stanley conjectured that the Jack ...
Ryan Mickler's user avatar
  • 375
2 votes
1 answer
255 views

I recently discovered a recursive, closed-form summation formula that appears to compute the complete homogeneous symmetric polynomial $h_n(x_0, x_1, \dots, x_{m-1})$, but in a more structured and ...
user avatar
3 votes
1 answer
122 views

Suppose real-valued wave function $\psi(r_1,\dots,r_N)\in L^2$ (or $H^1$) is unnormalized and antisymmetric, that is: $$\psi(r_{\sigma(1)},\dots,r_{\sigma(N)}) = \text{sgn}(\sigma)\psi(r_1,\dots,r_N),\...
janskel's user avatar
  • 39
2 votes
0 answers
116 views

In this paper, On Hall-Littlewood polynomials, the author states that traditionally Hall-Littlewood polynomials been interpreted in terms of point counting over finite fields. My question is what ...
user236626's user avatar
  • 233
2 votes
2 answers
250 views

For positive integers $k,n$ with $k\le n{-}1$ and $a\in\mathbb{R}^{n}$, let $S_{k}(a):=\sum_{i_{1}<\dots<i_{k}}a_{i_{1}}\dotsm a_{i_{k}}$ be the $k$-th elementary symmetric polynomial. If the ...
Annemarie Kästner's user avatar
  • 356
2 votes
0 answers
73 views

Let $\lambda$ and $\mu$ be weights of some simply-connected reductive group $G$ over $\mathbb{C}$, viewed as elements of the affine Weyl group of $G$. Then one of the many ways to define the ...
Stefan Dawydiak's user avatar
  • 513
10 votes
1 answer
379 views

Let $h_r$ and $e_r$ be the complete and elementary symmetric functions of degree $r \in \mathbb{N}_0$ and let $s_\lambda$ be the Schur function labelled by the partition $\lambda$. Let $e_r^\bot$ ...
Mark Wildon's user avatar
  • 11.9k
3 votes
0 answers
248 views

I'm interested in Young Tableaux filled with numbers from $1$ to $n$ with values increasing in every row and decreasing in every column (English notation). Is there some formula for the number of such ...
Fedor Ushakov's user avatar
  • 31
8 votes
0 answers
296 views

I am wondering whether there is a (concise?) combinatorial proof of the following identity. Let $\lambda$ be a partition of $n$. Then we have \begin{equation} \label{eq:m=s} m_\lambda = \sum_{\alpha\...
Byung-Hak Hwang's user avatar
  • 81
8 votes
2 answers
357 views

We know that for a sequence of real numbers $(a_n)$, the fact that $\lim_n a_n=a\neq 0$ is "more or less" equivalent to saying that the generating function $\sum_n a_n t^n$ has radius of ...
Yifeng Huang's user avatar
  • 824
3 votes
1 answer
218 views

Consider $N$ variables, $\vec{v} = v_1, ... v_N$. This is an $N$-input, $N$-output problem. Let $\sigma \in \mathfrak{S}_N$, the permutation group, that permutes the order of these variables. A set ...
YKY's user avatar
  • 588
3 votes
0 answers
125 views

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 $m_{\rho}(\mathbf{z})$ ...
T. Amdeberhan's user avatar
  • 43.9k
11 votes
3 answers
429 views

The general question For $1\leq k\leq n$, let $$e_k(a_1,\dots,a_n):=\sum_{j_1<\dots<j_k}a_{j_1}\cdots a_{j_k}$$ be the $k$-th elementary symmetric polynomial. Let $a_1,\dots,a_n<1$ and $e_1(...
Annemarie Kästner's user avatar
  • 356
3 votes
1 answer
176 views

Let $F$ be a field, let $f \in F[x]$, let $E$ be the splitting field of $f$, and let $e \in E$ be written in terms of the roots of $f$. I'm looking for a simple way to establish if $e \in F$. If $E/F$ ...
en-drix's user avatar
  • 434
6 votes
1 answer
367 views

Let $n$ be a positive integer and consider the ring $R$ of power series over $\mathbb{Q}$ in commuting variables $x_1,y_1,x_2,y_2,...$. Let the symmetric group $\mathfrak{S}$ of permutations of the ...
Jeremy Martin's user avatar
  • 1,510
2 votes
0 answers
150 views

This question is a continuation of a question asked yesterday which had a very nice answer. Consider the summation $$\sum_{\lambda \subset (k)^n} \dim S_{\lambda^t} (\mathbb{C}^k) \cdot \dim S_{[\...
Rellek's user avatar
  • 563
6 votes
1 answer
275 views

Suppose $n$ and $k$ are two integers. Then I am interested in having a closed form for the sum $$\sum_{\lambda \subset k \times n} S_\lambda (\mathbb{C}^n),$$ where $S_\lambda$ denotes the Schur ...
Rellek's user avatar
  • 563
5 votes
2 answers
361 views

Edit: The problem I pose here is impossible to solve with the basis $H$, in the answer I made to this post I explain why. The only way I can think it to amend the situation would be to try with ...
P.Luis's user avatar
  • 161
13 votes
2 answers
3k views

As mentioned in the post on Stanley's 25 positivity problems, Tatsuyuki Hikita posted a preprint on October 16, 2024 purporting to prove Problem 21, the Stanley–Stembridge conjecture about e-...
Joshua P. Swanson's user avatar
  • 1,022
8 votes
0 answers
178 views

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $H^k$ be the space of homogeneous degree $k$ harmonic polynomials in $\mathrm{Sym}(\mathfrak{g}^*)$ and $H\subset\mathrm{Sym}(\mathfrak{g}^*)...
Stefan Dawydiak's user avatar
  • 513
2 votes
0 answers
101 views

For some calculations related to the unramified principal series of ${\rm GL}(n)$ over a $p$-adic field, I need to compute Hall-Littlewood polynomials that are associated to $n$-tuples that are not ...
Paul Broussous's user avatar
  • 6,459
6 votes
1 answer
460 views

Let $P$ and $Q$ be polynomials over $\mathbb C$, and $n\in\mathbb N$ be a positive integer. I'm interested in the root sums of the form $$ \sum_{P(x)=0}\frac{Q(x)}{P'(x)^n},$$ where the sum runs over ...
El Rafu's user avatar
  • 99
16 votes
1 answer
807 views

Let $A_n$ denote the set of noncrossing fixed point free involutions in the symmetric group $S_{2n}$. "Noncrossing" means that if $a<b<c<d$, then not both $(a,c)$ and $(b,d)$ can be ...
Richard Stanley's user avatar
  • 53.7k
5 votes
0 answers
134 views

In Gian-Carlo Rota's article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations", at the end of the third lesson he states a theorem: "Every differential ...
Ryan's user avatar
  • 226
10 votes
1 answer
268 views

Let $a(n)$ be A225114 (i.e., number of skew partitions of $n$ whose diagrams have no empty rows and columns). Let $b(n)$ be an integer sequence with generating function $B(x)$ such that $$ B(x) = \...
user avatar
1 vote
1 answer
142 views

Let $n,m,k$ be positive integers. Consider the action of symmetric group $S^n$ on $\mathbb{R}^{n\times i}$ (for $i\in \{m,k\}$) by permuting rows; i.e. for each $\pi\in S^n$ and every $n\times i$ ...
AB_IM's user avatar
  • 4,842
7 votes
1 answer
424 views

If $\lambda$ is a partition with at most $n$ parts, let $s_\lambda$ be the corresponding Schur polynomial in $n$ variables $x_1,\ldots,x_n$. In particular, for $a \geq 0$, $s_{a}$ is the complete ...
Evan O'Dorney's user avatar
  • 1,282
5 votes
0 answers
240 views

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 ...
Nathaniel Johnston's user avatar
  • 6,882
0 votes
0 answers
218 views

Let $f,g \in \mathbb{C}[x,y]$ with total degrees $\deg_{1,1}(f),\deg_{1,1}(g) \geq 1$. Write, $f=a_ny^n+a_{n-1}y^{n-1}+\cdots+a_1y^1+a_0$ and $g=b_my^m+b_{m-1}y^{m-1}+\cdots+b_1y^1+b_0$, for some $n,m ...
user237522's user avatar
  • 2,883
2 votes
0 answers
210 views

The monomial symmetric polynomials are defined see Wikipedia. For an arbitrary partition $\lambda$ with $n$ parts I'm trying to find the following values: $$m_{\lambda}(\omega_0,\dotsc,\omega_{n-1})$$ ...
wkmath's user avatar
  • 63
10 votes
0 answers
224 views

This question by Rellek reminds me of the following problem. Alan Sokal has conjectured that if we replace the elementary symmetric function $e_k$ in the dual Jacobi-Trudi matrix for $s_\lambda$ (a ...
Richard Stanley's user avatar
  • 53.7k
0 votes
0 answers
105 views

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(...
Terence C's user avatar
  • 161
4 votes
1 answer
179 views

Let $h_i (x)$ denote the complete symmetric function of degree $i$ in some set of variables $x = (x_1 , x_2 , \dots)$. Then the minors of the Toeplitz matrix $T (x) = \left(h_{i-j} (x) \right)_{i,j}$ ...
Rellek's user avatar
  • 563
6 votes
2 answers
1k views

Quite surprisingly the following question appears while studying the billiard dynamics. Assume we have $2n$ real numbers: $ x_1, x_2,..., x_{2n}$. Assume also that $S_k=0$ for any odd positive integer ...
Dmitri Scheglov's user avatar
  • 678
0 votes
0 answers
132 views

Let's say that a Schur polynomial is degenerate if its number of variables is less than the weight of the partition it is associated to. For example, according to Sage, the Schur polynomial of the ...
Stéphane Laurent's user avatar
  • 2,590
11 votes
2 answers
365 views

Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ ...
Greg Zitelli's user avatar
  • 1,194
2 votes
0 answers
124 views

Let $B$ be the unit ball in $\mathbb{R}^N$. Denote by $\|\cdot\|$ the usual norm in $H^1_0(B)$. In (2) of the paper is said that $$ |u(x)| \leq \frac{\|u\|}{|x|^{(N-2)/2}}, $$ for any radial function $...
ThiagoGM's user avatar
  • 239
0 votes
1 answer
152 views

Consider $\Omega\subset \mathbb{R}^N$ a bounded domain. By definition, the space of radial functions in $H^1_0(\Omega)$ is $$ H^1_{0, \text{rad}}(\Omega) = \{u \in H^1_0(\Omega) : u = u \circ R, \...
ThiagoGM's user avatar
  • 239
6 votes
1 answer
317 views

For the complex ball $|z|^2\le 1$ in $\mathbb{C}^n$, there is a Poisson kernel proportional to $|x-z|^{-2n}$. This is generalized to the unitary group $U(N)$ so that in the complex matrix ball $Z^\...
thedude's user avatar
  • 1,549
7 votes
0 answers
176 views

Asked initially at MSE. I would like to know how to express the Fourier series of a symmetric function, $f(\theta_1,...,\theta_N)$, in terms of Schur polynomials $s_\lambda(x_1,...,x_N)$ in the ...
thedude's user avatar
  • 1,549
3 votes
0 answers
112 views

Let $\widehat{\cal H}_n$ be the type A degenerate affine Hecke algebra on $n$ strands, and let $x_1,\cdots,x_n$ be the dots. Inside of this algebra lies the algebra $\mathbb C S_n$, and the Young ...
Fan Zhou's user avatar
  • 311
19 votes
1 answer
773 views

Suppose that $n$ hamsters are at the points $(1,n)$, $(2,n),\dots, (n,n)$ in the plane. They walk independently one step east with probability $p$ or one step south with probability $1-p$, until ...
Richard Stanley's user avatar
  • 53.7k
3 votes
1 answer
158 views

The Poisson kernel of the unitary group is $$ P(Z,U)=\frac{\det(1-ZZ^\dagger)^N}{\det(1-ZU^\dagger)^N\det(1-UZ^\dagger)^N}.$$ It has a reproducing property, $\int dU P(Z,U)f(U)=f(Z)$, where $dU$ is ...
Marcel's user avatar
  • 2,572
3 votes
0 answers
134 views

Let $\mathfrak S_n$ be the symmetric group of permutations of $n$ letters and let $S = \sum_{\sigma\in\mathfrak S_n} \sigma$ be the symmetrization operator. Let $\Lambda_n^r$ be the vector space of ...
Martin Rubey's user avatar
  • 5,942
1 vote
0 answers
215 views

Let $\chi$ be a rational character of $G:=S_n$, and we want to know whether it decomposes into irreducibles $\chi_\lambda$, for $\lambda\in\Lambda$, with $\Lambda$ given, as $$ \chi=\sum_{\lambda\in\...
Dima Pasechnik's user avatar
  • 14.7k
0 votes
0 answers
121 views

Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
pallab1234's user avatar
  • 397
3 votes
1 answer
495 views

I'm not familiar with the formalism of plethysm, so I need some help in unpacking the plethystic substitution $h_n[n\mathbf{z}]$ found in eqns. 5.6 and 5.9 of "Combinatorics of labelled ...
Tom Copeland's user avatar
  • 11.1k
3 votes
1 answer
329 views

Let $G:=\operatorname{GL}_{N}$ act on its Lie algebra $\mathfrak{g}:=\mathfrak{gl}_{N}$ by conjugation. Then it acts naturally on the associated ring $\mathcal{O}(\mathfrak{g})$ of (algebraic or ...
Shaul Zemel's user avatar
  • 161

1
2 3 4 5
…
8