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 ...
0
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4
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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 ...
1
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1
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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 $$p^*_k=\sum_{i=1} \left((x_i-i+1/2)^k-(-i+1/2)^k\...
5
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0
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Interpretation of Hilbert/Frobenius series shift
Let $V = \oplus_{i\geq 0} V^i$ be a graded vector space.
Recall that the Hilbert series is defined as
$$F(q) = \sum_{i\geq 0} q^i \operatorname{dim}(V^i),$$
or if we have a graded $S_n$-module, $M$, ...
40
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3
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Characterizing positivity of formal group laws
The formal group law associated with a generating function $f(x) = x + \sum_{n=2}^\infty a_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my thesis, I found a large number of examples of ...
0
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0
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148
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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$, $...
4
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1
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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}$...
4
votes
1
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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{...
3
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1
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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 ...
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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 ...
3
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2
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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 ...
14
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1
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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)|$$ where $...
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Natural maps between Schur functors: understanding the image
Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map
$$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$
Let $[\Lambda^2 V]...
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Holonomic generating function
Let $\lambda$ denote a hook of size $d$ and $c(\Box)$ the content of $ \Box \in \lambda $. Let $ \text{Hooks}(d) $ be the set of hooks with $d$ boxes. Define
\begin{align}
B(d)&=
\frac{1}{d!h^{d-1}...
6
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1
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438
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Rational generating function and recursion
Let $\lambda$ denote a hook of size $d$ and $c(\Box)$ the content of $ \Box \in \lambda $. Let $ \text{Hooks}(d) $ be the set of hooks with $d$ boxes. Define
\begin{align}
B(d)&=
\frac{1}{d!} \...
5
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0
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998
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Internal tensor product of strict polynomial functors: is there a more explicit definition?
In the paper Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors, arXiv:1203.0311v4, Krause defines something that he calls an "internal tensor product" on the category of ...
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1
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What is currently known or conjectured about q,t-Kostka polynomials?
The $q,t$-Kostka polynomials $K_{\lambda,\mu}(q,t)$ appear as the change of basis coefficients between Macdonald polynomials $H_\mu(x;q,t)$ and Schur functions $s_\lambda(x)$:
$$H_\mu(x;q,t)=\sum_{\...
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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-...
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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 $\...
14
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3
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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,...
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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
$$H_k=\Q[...
3
votes
1
answer
550
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Product of Schur functions
Given two sets of variables $X=\{x_1,\cdots,x_n\}$, $Y=\{y_1,\cdots,y_m\}$, and two partitions $\lambda$ and $\mu$. Is there a formula for the product of the Schur functions $s_{\lambda}(X) s_{\mu}(Y)$...
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5
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Understanding a quip from Gian-Carlo Rota
In the chapter "A Mathematician's Gossip" of his renowned Indiscrete Thoughts, Rota launches into a diatribe concerning the "replete injustice" of misplaced credit and "forgetful hero-worshiping" of ...
3
votes
1
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Cauchy identity, with sum restricted over partitions with first part $\leq n$
The Cauchy Identity
$$ \sum_{\nu}s_{\nu}(x)s_{\nu}(y) = \prod_{j,k=1}^{\infty}\frac{1}{1-x_{j}y_{k}} $$
expresses the sum over all integer partitions of the product of pairs of Schur polynomials as ...
10
votes
1
answer
494
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Cauchy identity in three sets of variables?
The Cauchy identity states that
$$
\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 ...
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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 ...
2
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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 ...
8
votes
1
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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_\...
2
votes
1
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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}...
3
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1
answer
494
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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 ...
2
votes
1
answer
411
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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 ...
25
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1
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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}$$
be the $k$-th elementary symmetric function of $\tfrac1{1},\tfrac1{2}, \ldots, \...
1
vote
5
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Inequality with symmetric polynomials [closed]
How to prove the inequality $a^6+b^6 \geqslant ab^5+a^5b$ for all $a, b \in \mathbb R$?
3
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0
answers
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On finding simpler symmetries to differential equations
I have developed a differential equation for the variation of a star's semi-major axis with respect to its eccentricity.
It is as follows:
$$\frac{dy}{dx}=\frac{12}{19}\frac{y\left(1+\left(\frac{73}{...
1
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0
answers
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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 $...
6
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0
answers
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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 $$P_\lambda(x_1,\ldots,x_n;t)=\frac1{W_\lambda(t)}\sum_{w\in W}...
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Further aspects of a Hankel matrix of involution numbers
We have two conjectured generalizations of the question asked at
a Hankel matrix of involution numbers
by Tewodros Amdeberhan. Let $n!!=1!\,2!\cdots n!$.
Conjecture 1. Let $I_k$ denote the number of ...
1
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2
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Rank of a symmetric ideal
Let $\Sigma_m$ be the permutation group on $m$ letters and $R=\mathbb{C}[x_1,x_2,\dots,x_m]$. Let $\Sigma_m$ act on $R$ in the usual way. Let $R^{\Sigma_m}$ denote the ring of $\Sigma_m$ invariant ...
7
votes
1
answer
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Sum of the ratios of Schur functions
There are simple expressions for the sums of linear and quadratic combinations of Schur functions over all partitions (including the empty one)
$$
\sum_\lambda s_\lambda(x)=\prod_{i}\frac{1}{1-x_i}\...
3
votes
0
answers
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$S_n$ 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 &...
10
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1
answer
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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 ...
2
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0
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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_{\...
5
votes
1
answer
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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 ...
15
votes
2
answers
791
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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 $\mathcal{L}:R\rightarrow R$...
12
votes
3
answers
442
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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 ...
14
votes
1
answer
611
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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, ...
13
votes
1
answer
419
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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 ...
6
votes
2
answers
243
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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^{-...
4
votes
1
answer
312
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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 ...
3
votes
1
answer
255
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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,...