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3
votes
0answers
284 views

On a positivity property of Hall-Littlewood polynomials

Here's the new, more thought through version. Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is ...
3
votes
1answer
47 views

Counting a Modified Class of Standard Young Tableau

Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be a partition, with $|\lambda|:=N$. Attach an extra box to $\lambda$ to the right end of the $r$'th row. In coordinate form, the last box on row $r$ has ...
2
votes
0answers
141 views

computation with Hilbert scheme of $n$ points on $\mathbb C^2$ [closed]

How can we show that $$\sum_{n = 0}^\infty q^n \operatorname{char}_T S^n(\mathbb C[x,y])= \prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$ where $\operatorname{char}_T V$ denotes the character ...
10
votes
3answers
634 views

A class of matrix determinants between Wronskians and Vandermondes

Update: see below Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions ...
4
votes
1answer
150 views

Shift-invariant symmetric functions in representation theory?

The connection between symmetric functions and representation theory is well-known. Now consider the subspace of symmetric functions that are shift-invariant, that is, functions satisfying ...
3
votes
1answer
259 views

Comultiplication on Schur functors (& functions). Can it be seen from categorical perspective ?

Consider category of vector spaces. Consider functors from it to itself. They actually form an algebra - since vector spaces can be added and tensor multiplied. Question Is there co-product on this ...
4
votes
2answers
150 views

Isotypic components of the action of the symmetric group on polynomials

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S_n$. The isotypic component of the trivial representation is ...
9
votes
2answers
378 views

A particular specialization of symmetric polynomials: is it bijective?

Let $\Lambda^d_n$ the space of symmetric polynomials in $n$ variables, with maximum 'partial degree' of each variable $d$. A basis for this space is the set of symmetrized monomials $m_\lambda$, where ...
0
votes
0answers
16 views

Reference or a short argument that a certain subset generates the ring of p-typical symmetric functions under plethysm

Let p be a prime and$\Lambda_p$ be the subring of the ring of symmetric functions $\Lambda$ (over $\mathbb{Z}$) such that $$x \in \Lambda_p$$ iff there is an $i \in \mathbb{N}$ such that $p^ix \in ...
2
votes
0answers
114 views

Anti-arithmetic product of symmetric functions: (why) is it integral?

This is an analogue of MathOverflow question #138148. Indeed it is so analogous that I wrote the following by copypasting said question and making the necessary changes. For every commutative ring ...
2
votes
1answer
171 views

AM-GM interpolation in the limit

Given a sequence of positive numbers $X = (x_1, x_2, \dots)$, define the $k$th elementary symmetric mean of the first $n$ entries to be $$ S(X, n, k) := \frac{\displaystyle\sum_{1 \leq i_1 < ...
4
votes
1answer
204 views

Is the “renormalized third comultiplication” on $\mathbf{Symm}$ integral?

Background: For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means ...
1
vote
0answers
64 views

The relation on the set of functions

Let $\varphi: \mathbb{R}^{2} \to \mathbb{R}$ be a symmetric (not necessarily continuous) function (so, $\varphi(x,y)=\varphi(y,x)$ $\forall (x,y)\in \mathbb{R}^{2}$), let $\mathcal{F}$ be the set of ...
8
votes
2answers
251 views

How do I find coefficients of a product expansion

Any power series $f(t) = 1 + t \mathbb{Z}[[t]]$ can be uniquely expanded in the following two ways: $$1 + \sum_{i=1}^\infty f_i t^i = \prod_{i=1}^\infty (1-t^i)^{-n_i}$$ Here, the $f_i$ and $n_i$ ...
3
votes
1answer
280 views

An identity for elementary symmetric functions

Trying to understand a result in a representation theoretical paper, I realized that it implies the following elementary identity for symmetric functions. My question is whether this identity is true, ...
0
votes
0answers
39 views

Closed-for expression for Newton-Girard symmetric polynomials with 0/1 variables

There are $n$ Bernoulli $s_i\in\left\{0,1\right\}$, $i=1,...,n$ with equal marginals $\Pr(s_i=1)=\theta$ $\forall i$ so that E$(s_i)=\theta$. Their standardized mean deviations are \begin{equation*} ...
1
vote
1answer
154 views

A Simple Bijective Proof Of Stanley's Hook-Content Formula for Hook Shapes

this is my first post on math overflow so I hope it goes well. I believe I have a fairly simple bijective proof for Stanley's Hook-Content Formula in the case of hook shapes. I wanted to see if ...
6
votes
0answers
103 views

Littlewood-Richardson coefficients for Jack symmetric functions

Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(\alpha)$. We define a scalar product $\langle \cdot,\cdot\rangle_\alpha$ on $\Lambda$ by setting $\langle ...
5
votes
1answer
223 views

A formula on Kronecker coefficients

Accidentally, I proved the following formula for the Kronecker coefficients using some obscure method. $$g_{lm^n,mn^l}^{m^{nl}}=1,\ \forall l,m,n\in\mathbb{N},$$ where $n^m$ is the rectangle ...
4
votes
0answers
173 views

Littlewood-Richardson rule for Schubert polynomials

What is the current state of the problem of finding a combinatorial rule for multiplying two Schubert polynomials? Is the problem still open?
11
votes
1answer
309 views

Why are the power symmetric functions sums of hook Schur functions only?

One interesting fact in symmetric function theory is that the power symmetric function $p_n$ can be written as an alternating sum of hook Schur functions $s_{\lambda}$: $$ p_n = \sum_{k+\ell = n} ...
5
votes
0answers
164 views

Power sums and Jack symmetric functions

Let $\Lambda$ be the algebra of symmetric functions in infinitely many variables over $\mathbb{C}$. The $n$-th power sum symmetric function $p_n$ is defined (formally) as \begin{equation} p_n=\sum_i ...
10
votes
1answer
256 views

Reference request: Grothendieck groups of Hecke algebras at root of unity and symmetric functions

Let $\zeta$ be an $\ell^{\text{th}}$ root of unity, and consider $H_n(\zeta)$, the (finite) Hecke algebra of type A. One can consider a dual pair of Hopf algebras arising from this data, denoted ...
1
vote
1answer
340 views

Prove or disprove $ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > \int_{0}^{\infty} \int_{-\infty}^{-x} f(x)f(y)dydx. $

Consider a symmetric, unimodal distribution $f(x)$ such that $\int_{0}^{\infty} f(x) > 1/2$. I want to prove or disprove the following: $$ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > ...
9
votes
2answers
837 views

Can you find linear recurrence relation for dimensions of invariant tensors?

Let $V$ be a finite dimensional highest weight representation of a (semi)-simple Lie algebra. For each $n\ge 0$ take $a_n$ to be the dimension of the space of invariant tensors in $\otimes^n V$. In ...
2
votes
0answers
125 views

A generalization of Macdonald functions?

I am interested in finding a set of functions $f(z_1,\cdots ,z_k;q,\,t)$, conjecturally polynomials, which depend on two parameters $(q,t)$ and an integer $k$, and are orthogonal under the following ...
-1
votes
1answer
125 views

Algorithm to find symmetric function given specialization

I have a symmetric function f(c1,c2,c3,c4,c5) which, when c1 < c2 < c3 < c4 < c5, has the form p1(c1)+p2(c2)+p3(c3)+p4(c4)+p5(c5), where the p_i's happen to be polynomials of degree ...
1
vote
1answer
143 views

Roots of the derivative as symmetric functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
9
votes
4answers
2k views

Expressing power sum symmetric polynomials in terms of lower degree power sums

Is there an explicit formula expressing the power sum symmetric polynomials $$p_k(x_1,\ldots,x_N)=\sum\nolimits_{i=1}^N x_i^k = x_1^k+\cdots+x_N^k$$ of degree $k$ in $N < k$ variables entirely ...
2
votes
3answers
184 views

Constructing the sum of two pairs of symmetric polynomials

I have a question about generating a certain set of symmetric polynomials. I believe that what I'm looking for is known result, but I'm not 100% sure. Suppose that I have two sets of indeterminates ...
5
votes
1answer
150 views

Hopf algebra structure on the ring of quasisymmetric functions

I'm looking for a particular description of the Hopf algebra structure on the ring of quasisymmetric functions. Let me illustrate by giving this kind of description for the Hopf algebra of symmetric ...
8
votes
3answers
525 views

Arithmetic product of symmetric functions: why is it integral?

For every commutative ring $A$, let $\mathbf{Symm}_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}_{\mathbb{Z}}$. We can define a ...
6
votes
3answers
590 views

universality of Macdonald polynomials

I have been recently learning a lot about Macdonald polynomials, which have been shown to have probabilistic interpretations, more precisely the eigenfunctions of certain Markov chains on the ...
3
votes
1answer
108 views

Degree principles for non-symmetric polynomials

A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...
1
vote
0answers
61 views

Jack symmetric functions and their inner products

I have some questions regarding Jack polynomials. I use the notation of of I.G. Macdonald's book "Symmetric Functions and Hall polynomials". Let $\Lambda$ be the ring of symmetric functions over ...
1
vote
2answers
145 views

References request: representations of Heisenberg algebra.

Let $p_1, p_2, \ldots$, be the power sum symmetric functions. Let $p_n^* = n \frac{\partial}{\partial p_n}$. Then $$ p_n^* p_m - p_m p_n^* = \delta_{m, n} 1. $$ Where could I find this result in some ...
8
votes
1answer
291 views

Plethysm of $\mathrm{QSym}$ into $\mathrm{QSym}$: can it be defined?

I will denote by $\Lambda$ the ring of symmetric functions, and by $\mathrm{QSym}$ the ring of quasisymmetric functions (both in infinitely many variables $x_1$, $x_2$, $x_3$, ..., both over $\mathbb ...
4
votes
2answers
258 views

Cyclically symmetric functions

Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)? E.g., do the functions $x+y+z$, $xy+yz+zx$, and $x^2y+y^2z+z^2x$ generate the ...
4
votes
1answer
314 views

Criteria for ghost-Witt vectors: looking for history and references

I am looking for references (both of the readable and of the historical kind!) for the following result (which I formulate in one of its least general forms, so as not to complicate the discussion). I ...
2
votes
1answer
231 views

Other Variant of Schur Polynomials/Functions

We know that an important variant of the Schur Polynomial is the shifted Schur Polynomials that was developed by Okounkov & Olshanski. The question here is: is there any other variant of Schur ...
4
votes
1answer
269 views

To compute minors of Jacobian of symmetric polynomials

For any $n$ tuple $f_1,f_2,\dots,f_n$ in the polynomial ring $\mathbb{C}[x_1,x_2,\dots,x_n]$ one has Jacobian, expressed by the $(n \times n)$-determinants: $$ ...
6
votes
2answers
496 views

Expression for the sum of square roots of zeros of a polynomial

Let $f(x)$ be a polynomial of degree $n$ with rational coefficients whose zeroes are nonnegative real numbers: $x_1, \dots, x_n\geq 0$. General question. Does there exist a simple expression for the ...
4
votes
1answer
246 views

A nice generating set for the symmetric power of an algebra

I'm looking for a reference for the following fact. Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic zero. Let $S^n(A)$ ...
5
votes
1answer
225 views

A “known” Pythagorean identity in algebra?

Some will recognize this as similar to a question I asked before, but I want to ask it without the trigonometry. Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $x_1,x_2,x_3,\ldots$. ...
3
votes
0answers
72 views

Is the quasisymmetric expansion of the inner product of two Schur functions known?

The question is in the title, however; there is a Hopf algebra of quasisymmetric functions which has the Hopf algebra of symmetric functions as a sub - Hopf algebra. The quasisymmetric functions have ...
2
votes
1answer
187 views

S_n invariants in a free associative algebra (“noncommutative symmetric polynoms”)

What is known about S_n group invariants in a free associative (noncommutative) algebra k < x_1, ...x_n >) ? (S_n natural acts by permutations on generators). What is Poincare series ? Is it ...
5
votes
2answers
425 views

Can one recognize this symmetric function?

$\newcommand{\lm}{\lambda}$ $\newcommand{\bR}{\mathbb{R}}$ Let $m$ be an integer $>1$. Define $$ I_m:\bR^m\to \bR,\;\; I_m(\lm_1,\dotsc, \lm_m)=\int_{S^{m-1}}\exp\Bigl(-\sum_{j=1}^m ...
1
vote
1answer
230 views

Majorization of power sum symmetric functions

I'm wondering if there is a characterization for whether $$p_\lambda(x_1, \dotsc, x_r) \geq p_\mu (x_1, \dotsc, x_r) \text{ for every $r \in \mathbb{N}$ and $x_1, \dotsc, x_r \in \mathbb{R}^+$}.$$ (Or ...
1
vote
0answers
159 views

Generalized Schur polynomial from block Toeplitz matrices

By using the Jacobi-Trudi identity, one may interpret banded Toeplitz matrices, and minors of such matrices in terms of Schur polynomials, see for example ...
6
votes
2answers
750 views

Diagonal invariants of the symmetric group on $k[X_1,X_2,…,X_n,Y_1,Y_2,…,Y_n]$

This sounds like something that must have been answered long ago, but for some reason I can find nothing on it in the internet. (There has been lots of recent activity in diagonal covariants, related ...