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17
votes
2answers
1k views

What role does Cauchy's determinant identity play in combinatorics?

When studying representation theory, special functions or various other topics one is very likely to encounter the following identity at some point: $$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j ...
16
votes
0answers
242 views

Deforming a basis of a polynomial ring

The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions ...
13
votes
1answer
426 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} ...
12
votes
1answer
642 views

Can the Jacobi-Trudi identity be understood as a BGG resolution?

The thought process that led me to this question is that the identity $$ \left(\prod_i \frac1{1-x_i}\right)\left(\prod_i {1-x_i}\right)=1$$ can be understood as expressing exactness of the Koszul ...
11
votes
2answers
923 views

Sym(V ⊕ ∧² V) isomorphic to direct sum of all Schur functors of V

Let $V$ be a finite-dimensional $K$-vector space. Then, the symmetric power $\mathrm{Sym}\left(V\oplus \wedge^2 V\right)$ is isomorphic to the direct sum of all Schur functors applied to $V$ (each one ...
10
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 ...
10
votes
3answers
672 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 ...
10
votes
1answer
283 views

Dynamics of RSK

There is a way of viewing the RSK correspondence as a map (in fact, bijection) $A \overset{RSK}\longrightarrow \widehat{A}$ from $n\times n$ matrices with entries $\mathbb{N}$ to (weak) reverse plane ...
10
votes
1answer
272 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 ...
9
votes
2answers
854 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 ...
9
votes
2answers
405 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 ...
8
votes
3answers
555 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 ...
8
votes
2answers
274 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 $n_i$ ...
8
votes
4answers
495 views

Heisenberg algebra from Pieri operators and their transposes?

Let $Symm$ be the vector space with basis $(b_\lambda)$ given by the set of all partitions $\lambda$ (of all natural numbers), thought of as Young diagrams. Let $e_i$ be the degree $i$ Pieri operator ...
8
votes
2answers
438 views

Symmetric functions in type B and type D

It is well known that the symmetric groups have a very nice and explicit representation theory. This is in particular true when one works with the collection of all symmetric groups simultaneously, in ...
8
votes
3answers
577 views

$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th elementary symmetric polynomial of $n$ variables. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...
8
votes
1answer
335 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 ...
8
votes
1answer
272 views

Restriction of characters of hyperoctahedral groups.

The hyperoctahedral group $H_n$ has several descriptions; as a wreath product; as signed permutation matrices; as the Weyl group of type $B_n$ or $C_n$. In all these descriptions it is apparent that ...
8
votes
2answers
711 views

What is known about zero-sets of Schur polynomials?

Consider a set S of partitions not containing the empty partition (I would be happy with, say, all the partitions of size less than k, except for the empty one). Let $U_\lambda^{(r)}$ be the ...
8
votes
1answer
668 views

When the splitting fields of shifted generic polynomials are linearly disjoint?

Let me start by rigorously pose my question. Let $K$ be an algebraically closed field of characteristic $2$, let $n$ be an even integer number, let $f(X) = X^n + T_1 X^{n-1} + \cdots + T_n$, be the ...
8
votes
0answers
167 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 ...
7
votes
5answers
842 views

Rearrangement-style inequality with lots of terms and little evidence

This is another of the problems I designed for contests but wasn't able to solve on my own for years. Maybe AoPS is a better place for it, but let me try it here. [UPDATE: I have streamlined the ...
7
votes
1answer
424 views

Characterizing intersection of zero sets of elementary symmetric polynomials on R^n

Stated simply, the question is: Consider two elementary symmetric polynomials $\sigma_{k}$ and $\sigma_{k+1}$ on $\mathbb{R}^{n}$ with zero sets $U_{k}$ and $U_{k+1}$. Let $V_{i_{1}i_{2}\dotsb ...
7
votes
0answers
207 views

Deformation of the product of symmetric functions

Matthias Lederer and I are studying a deformation of the Littlewood-Richardson product of Schur functions. It's a bit complicated to define (and work in progress) so I won't give the full definition ...
6
votes
1answer
991 views

Jack polynomials as determinants

Jack symmetric polynomials are known to be generalizations of Schur functions $\chi_\lambda$, for which powerful Weyl determinant formulas are known. Are there any generalizations of two determinant ...
6
votes
2answers
785 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 ...
6
votes
1answer
958 views

Sum of products of p-th powers of roots of 1 and monomial symmetric functions

Hello mathematicians, i'm looking for explicit computations of expressions like $$ \sum_{\substack{0\leq i,j,k<n\\i\neq j\neq k \neq i}}\zeta_n^{ip^{k_1}+jp^{k_2}+kp^{k_3}} $$ and its ...
6
votes
3answers
629 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 ...
6
votes
2answers
564 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 ...
6
votes
0answers
147 views

Bijective proof for bialgebra identity of LR coefficients

Let $\Lambda$ be the ring of symmetric functions in infinitely many variables, $x_1$, $x_2$, .... For $f \in \Lambda$, let $\Delta(f) \in \Lambda \otimes \Lambda$ be $f(x_1 \otimes 1, 1 \otimes x_1, ...
5
votes
2answers
457 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 ...
5
votes
1answer
321 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 ...
5
votes
1answer
235 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 $x_1,x_2,x_3,\ldots$. ...
5
votes
1answer
337 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 ...
5
votes
1answer
296 views

schur weyl duality for real orthogonal groups and relation to hyperoctahedral groups

I am wondering whether the Lie groups $SO(n)$ and the hyperoctahedral groups $H_n$ form some sort of duality. I am mainly interested in how to parametrize the conjugacy classes of $H_n$ in terms of ...
5
votes
1answer
685 views

Symmetric polynomials theorem

Hello all, I would appreciate comments on the following question: A main theorem of symmetric functions might be formulated: Let k be a field of char. 0. Then $k[x_1,...,x_n]^{S_n} = ...
5
votes
1answer
695 views

Specializations of Schur functions at consecutive integers

Given a partition λ = (λ1, λ2, ..., λn) denote with sλ the associated Schur function. There exists a nice product formula for the principal specializations: ...
5
votes
1answer
157 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 ...
5
votes
1answer
219 views

Dimension of the span of all partial derivatives of a given symmetric polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below. Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set $$ ...
5
votes
0answers
58 views

Uniform generation of Symmetric Plane Partitons

In the conclusion of An Involution Principle-Free Proof of Stanley's Hook-Content Formula Krattenthaler notes that the techniques of the paper might be useful for finding bijective proofs of the ...
5
votes
0answers
185 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 ...
5
votes
0answers
414 views

Generalizations of generators / hyperplanes descriptions for cones to partially-ordered fields?

Background: given a finite-dimensional real vector space V of dimension d, I can define a pointed cone in two ways: either as a set of the form $\{r_1v_1 + \cdots + r_nv_n \mid r_1, \dots, r_n \in ...
4
votes
2answers
247 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 ...
4
votes
1answer
261 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 $S^n(A)$ ...
4
votes
1answer
165 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 ...
4
votes
1answer
484 views

Are the Schur functions the minimal basis of the ring of symmetric functions with the following properties?

Let $\Lambda$ denote the ring of symmetric functions in variables $x_1,x_2,\dots$ and with coefficients in $\mathbf{Q}$. Then $\Lambda$ is freely generated as an $\mathbf{Q}$-algebra by ...
4
votes
2answers
315 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
221 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 ...
4
votes
1answer
294 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: $$ ...
4
votes
0answers
203 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?