Questions tagged [symmetric-functions]
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.
370 questions
10 votes
3 answers
423 views
A question on the plethysm of complete symmetric functions
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 $n=2$. ...
- 419
1 vote
0 answers
136 views
Fibers of a mapping related to the discriminant
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 $\...
- 111
4 votes
0 answers
231 views
Is this operation on Stanley Diagrams known?
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 ...
- 375
2 votes
1 answer
215 views
A structured recursive formula for the complete homogeneous symmetric polynomial [closed]
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 ...
user562031
3 votes
1 answer
122 views
Characterization of the probability density induced by an antisymmetric wave function
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),\...
- 39
2 votes
0 answers
111 views
Hall-Littlewood polynomials as point counting
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 ...
- 203
2 votes
2 answers
243 views
Inequality of elementary symmetric polynomials written in terms of derivatives of $(s+ta_1)\dotsm(s+ta_n)$
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 ...
2 votes
0 answers
66 views
linear recurrence for Kostka-Foulkes polynomials
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 ...
- 513
10 votes
1 answer
377 views
Bernstein operator on Schur functions
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$ ...
- 11.7k
3 votes
0 answers
243 views
Antistandard Young Tableaux
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 ...
8 votes
0 answers
269 views
Schur expansion of monomial symmetric functions
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\...
8 votes
2 answers
349 views
Representation stability formulated as rationality in generating function of symmetric functions?
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 ...
- 824
3 votes
1 answer
193 views
How to build equivariant functions from invariant functions or vice versa? (for permutation groups)
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 ...
- 588
3 votes
0 answers
123 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})$ ...
- 43.7k
11 votes
3 answers
415 views
Positivity of elementary symmetric polynomials under linear fractional transformations
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}$$ be the $k$-th elementary symmetric polynomial. Let $a_1,\dots,a_n<1$ and $e_1(...
3 votes
1 answer
172 views
Is there a (simple) criterion for membership to the base field of an inseparable extension?
Let $F$ be a field, let $f \in F[x]$, let $E$ be the splitting field of $f$, and let $e \in E$ be written in terms of the roots of $f$. I'm looking for a simple way to establish if $e \in F$. If $E/F$ ...
- 424
6 votes
1 answer
358 views
Diagonal analogue of symmetric functions
Let $n$ be a positive integer and consider the ring $R$ of power series over $\mathbb{Q}$ in commuting variables $x_1,y_1,x_2,y_2,...$. Let the symmetric group $\mathfrak{S}$ of permutations of the ...
- 1,510
2 votes
0 answers
146 views
Formula for sum involving products of (symplectic) Schur functions
This question is a continuation of a question asked yesterday which had a very nice answer. Consider the summation $$\sum_{\lambda \subset (k)^n} \dim S_{\lambda^t} (\mathbb{C}^k) \cdot \dim S_{[\...
- 563
6 votes
1 answer
262 views
Number of semistandard tableaux of all possible shapes fitting within some rectangle
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 ...
- 563
5 votes
2 answers
348 views
Expansion of key polynomials in terms of non-symmetric Hall-Littlewood polynomials and charge-like statistics
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 ...
- 161
14 votes
2 answers
2k views
Status of the Stanley–Stembridge conjecture
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-...
- 1,026
8 votes
0 answers
177 views
Asymptotics of generalized exponents of highest weight modules
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}^*)...
- 513
2 votes
0 answers
99 views
Hall-Littlewood polynomials for $n$-tuples that are not partitions
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 ...
- 6,459
6 votes
1 answer
436 views
Sum of derivative of polynomial over its simple roots
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 ...
- 99
15 votes
1 answer
710 views
Some questions related to meanders
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 ...
- 53.5k
5 votes
0 answers
130 views
Differential equations analogue of fundamental theorem of symmetric functions
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 ...
- 226
10 votes
1 answer
268 views
Generating function for A225114
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 $$ B(x) = \...
user231922
1 vote
1 answer
140 views
Representation of equivariant maps
Let $n,m,k$ be positive integers. Consider the action of symmetric group $S^n$ on $\mathbb{R}^{n\times i}$ (for $i\in \{m,k\}$) by permuting rows; i.e. for each $\pi\in S^n$ and every $n\times i$ ...
- 4,942
7 votes
1 answer
411 views
Jacobi-Trudi-like identity with dual characters
If $\lambda$ is a partition with at most $n$ parts, let $s_\lambda$ be the corresponding Schur polynomial in $n$ variables $x_1,\ldots,x_n$. In particular, for $a \geq 0$, $s_{a}$ is the complete ...
- 1,292
5 votes
0 answers
217 views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,882
0 votes
0 answers
216 views
$\mathbb{C}(x,f,g)=\mathbb{C}(x,y)$, with each pair of $\{f,g,x\}$ not generating $\mathbb{C}(x,y)$
Let $f,g \in \mathbb{C}[x,y]$ with total degrees $\deg_{1,1}(f),\deg_{1,1}(g) \geq 1$. Write, $f=a_ny^n+a_{n-1}y^{n-1}+\cdots+a_1y^1+a_0$ and $g=b_my^m+b_{m-1}y^{m-1}+\cdots+b_1y^1+b_0$, for some $n,m ...
- 2,875
2 votes
0 answers
206 views
Monomial symmetric polynomials evaluation at roots of unity
The monomial symmetric polynomials are defined see Wikipedia. For an arbitrary partition $\lambda$ with $n$ parts I'm trying to find the following values: $$m_{\lambda}(\omega_0,\dotsc,\omega_{n-1})$$ ...
- 63
10 votes
0 answers
219 views
An $m$-positivity conjecture related to bivariate Jacobi-Trudi matrices
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 ...
- 53.5k
0 votes
0 answers
103 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(...
- 161
4 votes
1 answer
176 views
Are the minors of this Hadamard product Schur positive?
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}$ ...
- 563
6 votes
2 answers
1k views
Question about the sum of odd powers equation
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 ...
- 628
0 votes
0 answers
129 views
"Degenerate" Schur polynomials
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 ...
- 2,590
11 votes
2 answers
359 views
Matrix invariants for simultaneous conjugation by a finite subgroup of $\textrm{GL}_n$
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$ ...
- 1,184
2 votes
0 answers
120 views
Radial functions in $H^1_0(B)$ and a Strauss type inequality
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}}, $$ for any radial function $...
- 239
0 votes
1 answer
148 views
Dense subspace of space of radial functions in $H^1_0(\Omega)$
Consider $\Omega\subset \mathbb{R}^N$ a bounded domain. By definition, the space of radial functions in $H^1_0(\Omega)$ is $$ H^1_{0, \text{rad}}(\Omega) = \{u \in H^1_0(\Omega) : u = u \circ R, \...
- 239
6 votes
1 answer
314 views
Poisson kernel for the orthogonal groups
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^\...
- 1,549
7 votes
0 answers
169 views
Relation between Fourier series and Schur polynomials
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 ...
- 1,549
3 votes
0 answers
112 views
How to multiply dots with Young idempotents in the degenerate affine Hecke algebra (type A)
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 ...
- 311
18 votes
1 answer
754 views
Simple proof that certain walks in the plane don't intersect
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 ...
- 53.5k
3 votes
1 answer
152 views
Does the reproducing property of the unitary group Poisson kernel require a multiple of the identity?
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 ...
- 2,572
3 votes
0 answers
130 views
Bijectivity of a linear map between symmetric polynomials of even degree
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 ...
- 5,922
1 vote
0 answers
211 views
Efficient decomposition algorithm for characters of symmetric groups
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\...
- 14.6k
0 votes
0 answers
116 views
Quick calculation of a symmetric product with two indices
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 ...
- 397
2 votes
1 answer
447 views
Unpacking the plethystic substitution $h_n[n\mathbf{z}]$ in a paper by Aval, Bergeron and Garsia
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 ...
- 10.9k
3 votes
1 answer
321 views
Tangent space of a GIT quotient of $GL_{N}$
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 ...
- 161