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.

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Explicit form of raising and lowering operators in spherical gl(n) DAHA

I am working with polynomial representations of spherical subalgebra of double affine Hecke algebra (DAHA) for $\mathfrak{gl}_n$. Let's call this algebra $\mathfrak{A}_n$ for short. Typically we think ...
Peter Koroteev's user avatar
0 votes
4 answers
422 views

Show that sets are equal

Let $X=\{x_1,x_2,...,x_n\}$ and $Y=\{y_1,y_2,...,y_n\}$ be sets over a finite field $F$ with $p=char(F)>2$. Assume $$x_1^k+x_2^k+...+x_n^k=y_1^k+y_2^k+...+y_n^k,\ 1\leq k\leq n$$ I wanna ...
DavitS's user avatar
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1 vote
1 answer
148 views

On two types of shifted symmetric power sums

In the ring of shifted symmetric functions $\Lambda^*$ there are many ways to generalize the symmetric power sums. First of all, we have the functions $$p^*_k=\sum_{i=1} \left((x_i-i+1/2)^k-(-i+1/2)^k\...
Jan-Willem van Ittersum's user avatar
5 votes
0 answers
82 views

Interpretation of Hilbert/Frobenius series shift

Let $V = \oplus_{i\geq 0} V^i$ be a graded vector space. Recall that the Hilbert series is defined as $$F(q) = \sum_{i\geq 0} q^i \operatorname{dim}(V^i),$$ or if we have a graded $S_n$-module, $M$, ...
Per Alexandersson's user avatar
40 votes
3 answers
1k views

Characterizing positivity of formal group laws

The formal group law associated with a generating function $f(x) = x + \sum_{n=2}^\infty a_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my thesis, I found a large number of examples of ...
Jair Taylor's user avatar
0 votes
0 answers
148 views

Methods to get Holonomic functions

Let $a_n$ be a holonomic sequence. By definition, that means there exists a linear differential equation of finite order which annihilates $F(x)$, where $F(x):=\sum a_n x^n$. Similarly let $b_n$, $...
GGT's user avatar
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4 votes
1 answer
753 views

Symmetric functions of eigenvalues

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function, which is symmetric under the action of the symmetric group (acting on $\mathbb{R}^n$ by permuting the variables). Let $M_{n\times n}$...
Matthias Ludewig's user avatar
4 votes
1 answer
320 views

Decompostion of hook schur function in terms of cauchy product of holonomic functions

Let $s_{\lambda}$ denote the schur function and $\lambda$ is the partion of an integer. The schur function written in power sum symmetric basis apper as following. $\chi$ denote the character. \begin{...
GGT's user avatar
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3 votes
1 answer
510 views

Generalized Newton Identities

I learnt a lot of new words (Hall-Littlewood, Jack and Macdonald polynomials) but unfortunately everything I dug up is written without a single example and I still don't know the answer to a very ...
Hauke Reddmann's user avatar
7 votes
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237 views

Characterizing $n$-exceptions of the ring of symmetric polynomials

(Also in Mathematics Stack Exchange: https://math.stackexchange.com/questions/2528000/characterizing-n-exceptions-on-the-ring-of-symmetric-polynomials) We say that an homogeneous symmetric polynomial ...
Hector Blandin's user avatar
3 votes
2 answers
371 views

Bounded Degree in Ring of Symmetric Functions

The Ring of Symmetric Functions over a commutative ring $R$, $\Lambda$, is the subring of the ring of formal power series $R[[x_1, x_2, \dots]]$ such that $f \in \Lambda$ if $f$ is invariant under ...
Apprentice Counter's user avatar
14 votes
1 answer
417 views

Inequalities on elementary symmetric polynomials

I have recently come across the following result. Let $0 < d \leq n$. Given any vector $x \in \mathbb{R}^n$ that satisfies $e_{d-1}(x) = 0$, show that $$|x_1 \cdots x_d| \leq |e_d(x)|$$ where $...
Nick R's user avatar
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Natural maps between Schur functors: understanding the image

Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map $$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$ Let $[\Lambda^2 V]...
Daniil Rudenko's user avatar
1 vote
0 answers
174 views

Holonomic generating function

Let $\lambda$ denote a hook of size $d$ and $c(\Box)$ the content of $ \Box \in \lambda $. Let $ \text{Hooks}(d) $ be the set of hooks with $d$ boxes. Define \begin{align} B(d)&= \frac{1}{d!h^{d-1}...
GGT's user avatar
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6 votes
1 answer
438 views

Rational generating function and recursion

Let $\lambda$ denote a hook of size $d$ and $c(\Box)$ the content of $ \Box \in \lambda $. Let $ \text{Hooks}(d) $ be the set of hooks with $d$ boxes. Define \begin{align} B(d)&= \frac{1}{d!} \...
GGT's user avatar
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5 votes
0 answers
998 views

Internal tensor product of strict polynomial functors: is there a more explicit definition?

In the paper Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors, arXiv:1203.0311v4, Krause defines something that he calls an "internal tensor product" on the category of ...
darij grinberg's user avatar
18 votes
1 answer
788 views

What is currently known or conjectured about q,t-Kostka polynomials?

The $q,t$-Kostka polynomials $K_{\lambda,\mu}(q,t)$ appear as the change of basis coefficients between Macdonald polynomials $H_\mu(x;q,t)$ and Schur functions $s_\lambda(x)$: $$H_\mu(x;q,t)=\sum_{\...
CatO Minor's user avatar
1 vote
0 answers
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Hook-content polynomial 2

Recently I have proven the following identity \begin{align} \sum_{\lambda\in \text{different hook of size d}} \frac{1}{d!} (-1)^{ht(\lambda)-1} \, \dim \lambda \, \prod_{\Box \in \lambda} \frac{1}{1-...
GGT's user avatar
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3 votes
0 answers
206 views

Hook-content polynomial

Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\...
GGT's user avatar
  • 685
14 votes
3 answers
579 views

Schubert calculus expressed in terms of the cotangent space of the Grassmannians

Let $T^*_{\mathbb{C}}(Gr_{n,r})$ denote the cotangent space of the Grassmannian of $r$-planes in $\mathbb{C}^n$. Moreover, let $\Lambda^\bullet$ denote the exterior algebra of $T^*_{\mathbb{C}}(Gr_{n,...
Han Jin Ma's user avatar
2 votes
0 answers
112 views

Partial Flag Varieties and Quotients of Symmetric Polynomials

$\def\Q{\mathbf Q}\DeclareMathOperator{\Gr}{Gr}$First, consider a Grassmannian $\Gr(k, N)$ of $k$-dimensional subspaces in an $N$-dimensional space. It is known that its cohomology ring is $$H_k=\Q[...
Bubaya's user avatar
  • 259
3 votes
1 answer
550 views

Product of Schur functions

Given two sets of variables $X=\{x_1,\cdots,x_n\}$, $Y=\{y_1,\cdots,y_m\}$, and two partitions $\lambda$ and $\mu$. Is there a formula for the product of the Schur functions $s_{\lambda}(X) s_{\mu}(Y)$...
Iman's user avatar
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35 votes
5 answers
5k views

Understanding a quip from Gian-Carlo Rota

In the chapter "A Mathematician's Gossip" of his renowned Indiscrete Thoughts, Rota launches into a diatribe concerning the "replete injustice" of misplaced credit and "forgetful hero-worshiping" of ...
3 votes
1 answer
284 views

Cauchy identity, with sum restricted over partitions with first part $\leq n$

The Cauchy Identity $$ \sum_{\nu}s_{\nu}(x)s_{\nu}(y) = \prod_{j,k=1}^{\infty}\frac{1}{1-x_{j}y_{k}} $$ expresses the sum over all integer partitions of the product of pairs of Schur polynomials as ...
D. Donnelly's user avatar
10 votes
1 answer
494 views

Cauchy identity in three sets of variables?

The Cauchy identity states that $$ \prod_{i,j} \frac{1}{1-x_i y_j} = \sum_\lambda s_\lambda(x) s_\lambda(y), $$ where $s_\lambda(x)$ is the Schur function. Is there a known decomposition of the ...
andrewBee's user avatar
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8 votes
1 answer
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A combinatorial expression of Hall-Littlewood polynomials

This is related to the question Hall-Littlewood functions and functions on the nilpotent cone, and arises in the construction of Coulomb branches of gauge theories. The motivation is explained at the ...
Hiraku Nakajima's user avatar
2 votes
0 answers
178 views

Macdonald polynomials: existence and specializations

I am reading Macdonald's Symmetric Functions and Orthogonal Polynomials lecture notes, and have several related questions: In both the type A case (chapter 1) and the general irreducible root system ...
Roger Van Peski's user avatar
8 votes
1 answer
1k views

Details about plethysm

I'm currently working on plethysm, i.e. the character of the composition $S^\lambda(S^\mu(V))$ of the Schur functors $S^\lambda$ and $S^\mu$ on a complex vector space $V$. We note this character $s_\...
eti902's user avatar
  • 835
2 votes
1 answer
175 views

special values of symmetric functions at powers of $\frac1j$

Let $e_n(x_1,x_2,x_3,\dots)$ denote the $n$-th elementary symmetric function in the infinite variables $x_1,x_2,x_3,\dots$. Let $u$ and $v$ be the roots of $z^2-6z+1=0$. Question. Let $x_j=\frac1{j^8}...
T. Amdeberhan's user avatar
3 votes
1 answer
494 views

Singular locus of zero set of elementary symmetric polynomial

Let $\sigma_{m, r}$ be the degree-$r$ elementary symmetric polynomial in $m$ variables. Let $X_{m, r}$ be the zero set of $\sigma_{m, r}$ and $S_{m, r}$ its singular locus. I.e., $S_{m,r}$ is the set ...
Izaak Meckler's user avatar
2 votes
1 answer
411 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 ...
T. Amdeberhan's user avatar
25 votes
1 answer
826 views

Only finitely many values of the symmetric functions of $1/1,1/2,\ldots,1/n$ are $2$-adic integers (?)

For integers $n \geq k \geq 1$ let $$H(n,k) := \sum_{1 \leq i_1 < \cdots < i_k \leq n} \frac1{i_1 \cdots i_k}$$ be the $k$-th elementary symmetric function of $\tfrac1{1},\tfrac1{2}, \ldots, \...
user avatar
1 vote
5 answers
757 views

Inequality with symmetric polynomials [closed]

How to prove the inequality $a^6+b^6 \geqslant ab^5+a^5b$ for all $a, b \in \mathbb R$?
Constantor's user avatar
3 votes
0 answers
187 views

On finding simpler symmetries to differential equations

I have developed a differential equation for the variation of a star's semi-major axis with respect to its eccentricity. It is as follows: $$\frac{dy}{dx}=\frac{12}{19}\frac{y\left(1+\left(\frac{73}{...
Spoilt Milk's user avatar
1 vote
0 answers
142 views

Clustering Properties of Jack Polynomials at negative rationals

I'm trying to confirm or deny the truthfulness of a statement. Also I'm asking about a reference for the proof of the statement if it is true. Following Feigin et al, we say a partition $\lambda$ is $...
Hamed's user avatar
  • 593
6 votes
0 answers
250 views

Branching rules for type B/C/D Hall-Littlewood polynomials

For a root system $\Phi$ of rank $n$ with Weyl group $W$ and a dominant integral weight $\lambda$ consider the Hall-Littlewood polynomial $$P_\lambda(x_1,\ldots,x_n;t)=\frac1{W_\lambda(t)}\sum_{w\in W}...
Igor Makhlin's user avatar
  • 3,493
11 votes
1 answer
313 views

Further aspects of a Hankel matrix of involution numbers

We have two conjectured generalizations of the question asked at a Hankel matrix of involution numbers by Tewodros Amdeberhan. Let $n!!=1!\,2!\cdots n!$. Conjecture 1. Let $I_k$ denote the number of ...
Richard Stanley's user avatar
1 vote
2 answers
165 views

Rank of a symmetric ideal

Let $\Sigma_m$ be the permutation group on $m$ letters and $R=\mathbb{C}[x_1,x_2,\dots,x_m]$. Let $\Sigma_m$ act on $R$ in the usual way. Let $R^{\Sigma_m}$ denote the ring of $\Sigma_m$ invariant ...
Evan Wilson's user avatar
7 votes
1 answer
218 views

Sum of the ratios of Schur functions

There are simple expressions for the sums of linear and quadratic combinations of Schur functions over all partitions (including the empty one) $$ \sum_\lambda s_\lambda(x)=\prod_{i}\frac{1}{1-x_i}\...
Sasha's user avatar
  • 1,343
3 votes
0 answers
221 views

$S_n$ action on the sequences of transpositions

It is well-known, that any element $\rho$ of the symmetric group $S_n$ with $n-p$ cycles admits a unique presentation as a product of a sequence of transpositions $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i &...
user79456's user avatar
  • 401
10 votes
1 answer
625 views

Coefficients of polynomials appearing in Newton identities relating symmetric polynomials to power sums

Let $$p_k(x_1,\ldots,x_n)=x_1^k+\cdots+x_n^k$$ and let $$e_k(x_1,\ldots,x_n)=\sum_{1\le i_1<i_2<\ldots<i_k\le n}x_{i_1}\cdots x_{i_k} $$ be the $k$'th symmetric polynomial. It is well known ...
Joe Silverman's user avatar
2 votes
0 answers
70 views

Is there a formula for skew Macdonald functions similar to Jacobi-Trudi identity?

For the special case of Schur functions, we have the Jacobi-Trudi identity which can expresses $s_{\lambda/\mu}$ as a determinant. I wonder if there is any similar formulas for the general case $P_{\...
xmchenhit's user avatar
  • 115
5 votes
1 answer
227 views

Jack polynomials and the Witt algebra

The symmetric Jack polynomials $J_n^{\alpha}(x_1,x_2,..,x_{n+1})$, a special subset of the symmetric Jack functions presented in Stanley's paper in equation a) on page 80, can be represented by the ...
Tom Copeland's user avatar
  • 9,937
15 votes
2 answers
791 views

An orbit of symmetric polynomials

Consider the ring of polynomials $R:=\mathbb{Z}[x_1,x_2,x_3]$. Define the operators $E, I:R\rightarrow R$ by $Ef(x_1,x_2,x_3)=f(x_1-1,x_2,x_3)$ and the identity $If=f$. Let $\mathcal{L}:R\rightarrow R$...
T. Amdeberhan's user avatar
12 votes
3 answers
442 views

Integer-valued power sums

Suppose I have a positive number $d \in \mathbb{R}$ and a sequence of numbers $a_n \in [0,d]$ for $n \in \mathbb{N}$ with the following properties $$ \sum_{i=1}^{\infty} a_i^r \in \mathbb{Z} $$ for ...
Ulrich Pennig's user avatar
14 votes
1 answer
611 views

A Schur positivity conjecture related to row and column permutations

The problem Counting cycles after permuting within rows and columns reminds me of the following unpublished conjecture of mine. Let $D$ be any finite planar diagram (in the sense of Young diagram, ...
Richard Stanley's user avatar
13 votes
1 answer
419 views

Counting higher-dimensional partitions with symmetric function theory

My coauthors and I are writing a (mostly expository) paper in which we construct the Specht module. Our proof that the Specht module is irreducible in characteristic zero implies the following ...
John Wiltshire-Gordon's user avatar
6 votes
2 answers
243 views

Symmetric functions arising from continuous-time Markov chains

I'm interested in the following symmetric functions $s_k: \mathbb{R}_+^{k}\mapsto\mathbb{R_+}$: \begin{align*} s_k(x_1,x_2,\dots, x_k)&= \int\limits_{0=t_0<t_1<\dots<t_{k-1}<t_k=1} e^{-...
James Martin's user avatar
  • 3,787
4 votes
1 answer
312 views

Principal Minors of the Resultant

Let $x_1, \ldots, x_n$ be variables, $e_n$ be the elementary symmetric polynomials. I will denote the discriminant by $$D_n(x_1, \ldots, x_n) = \prod_{i<j} (x_i - x_j)^2$$ And a generalized ...
Nick R's user avatar
  • 1,047
3 votes
1 answer
255 views

elementary symmetric function identity

I want to evaluate the following sum: $\sum_{\lambda}e_k(\lambda)$ where the sum is over all partitions $\lambda$ that fit in an $m$ by $n$ grid and where $e_k(\lambda):=e_k(\lambda_1,\lambda_2,\dots,...
user162496's user avatar

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