The symmetric-functions tag has no usage guidance.

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**1**answer

339 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

**2**answers

274 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 ...

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**2**answers

325 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 ...

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votes

**2**answers

580 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

**1**answer

262 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 $S^n(A)$ ...

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votes

**1**answer

243 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
$x_1,x_2,x_3,\ldots$. ...

**4**

votes

**1**answer

223 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 ...

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**0**answers

79 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 ...

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votes

**1**answer

195 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 ...

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votes

**2**answers

460 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 ...

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**1**answer

246 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 ...

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**0**answers

164 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
...

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**1**answer

298 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:
$$
...

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vote

**0**answers

289 views

### Passing from Regular sequence to Prime ideal, for power sum symmetric polynomial

Let $S=\mathbb{C}[x_1,x_2,x_3,x_4]$ be a polynomial ring. Let $p_i=x_1^i+\cdots+x_4^i$ be the power sum symmetric polynomial in $\mathbb{C}[x_1,x_2,x_3,x_4]$.
Let $I=(p_1,p_2)$ be an Ideal of ...

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**1**answer

249 views

### Regular sequence of power sum symmetric polynomials in polynomial ring.

Let $S=\mathbb{C}[x_1,\dots,x_n]$ be a polynomial ring and $p_a=x_1^a+\cdots+x_n^a$ be a power sum symmetric polynomial in $S$. Let $n \geq 3$.
Question: To show $p_m,p_{2m}, \dots,p_{nm}$ forms a ...

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**0**answers

278 views

### Generalization of Macdonald polynomials?

Let $G$ be a semi-simple group with maximal torus $T$ and Weyl group $W$. It looks like from some geometric considerations I can define a family $P_{\lambda,\alpha}(q,t,z)$ of
$W$-invariant ...

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**0**answers

173 views

### root system generalizations of Sekiguchi-Debiard (aka Laplace-Beltrami) operators

For the root system $A_n$, taking the limit $q = t^\alpha$ and $t \to 1$, and letting $Y = (t-1) X -1$ one obtains from the Macdonald operator the so-called Sekiguchi-Debiard operator:
$$D_\alpha(X) ...

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**1**answer

270 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 ...

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**0**answers

210 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 ...

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**2**answers

796 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 ...

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633 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 ...

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**0**answers

302 views

### analogues of power sum polynomials for symmetric Laurent polynomials

To deal with root systems of type B C D, one needs to understand symmetric Laurent polynomials $\Lambda$. I am wondering if the naive definition of power sum symmetric Laurent polynomials form a basis ...

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**1**answer

305 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 ...

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148 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, ...

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846 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 ...

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votes

**2**answers

304 views

### L^2 basis of class functions on a compact Lie group that are point-wise small

Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta_1, \ldots, \theta_k) \mapsto e^{i \sum_j c_j \theta_j}$ for ...

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414 views

### Nontrivial algorithm to check for polynomial symmetry?

Hi.
As is known, a polynomial $P \in K[x_1, \dots, x_n]$ is symmetric when permuting its variables always yields the same polynomial. This immediately yields an algorithm $O(n!)$ to check for ...

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votes

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192 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 ...

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**1**answer

292 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 ...

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**2**answers

721 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 ...

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votes

**1**answer

968 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 ...

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votes

**0**answers

212 views

### Picking $n$ so that certain Schur functors of the standard representation of $S_n$ are linearly independent

Let $V_n$ be the standard permutation representation of the symmetric group $S_n$, and let $\mathbb{S}_{\lambda}$ denote the Schur functor associated to the partition $\lambda$.
Let $\lambda$ range ...

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**4**answers

509 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 ...

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votes

**1**answer

426 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 ...

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votes

**2**answers

455 views

### Convolution on symmetric group Sn

I have question regarding convolution of functions (say
g and h) defined on Sn. In Fourier space this is equivalent to IFT(G.H),
where G = FT(g) and H = FT(h).
Fast Fourier transforms (Clausen's FFT) ...

**2**

votes

**1**answer

290 views

### Restriction from $\mathfrak{gl}_{2n}$ to $\mathfrak{sp}_{2n}$

Hi,
I am faced with a finite-dimensional representation $V$ of $\mathfrak{gl}_{2n}$, whose character I know. I know how to use this character to determine the irreducibles for $\mathfrak{gl}_{2n}$ ...

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votes

**2**answers

929 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 ...

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votes

**1**answer

361 views

### A conjecture of parallelogram inside convex and central symmetric curve

Assume Q is a convex central symmetric curve, whose area is $\displaystyle S$. The area of the maximum parallelogram inside Q is $\displaystyle S'$.
How to prove the conjecture that $\displaystyle ...

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votes

**1**answer

255 views

### some rational functions over a field of characteristic 2

I would like to know what are the formal power series $$f(t) = \sum_a \omega_a t^{-a}$$ over an algebraicially closed field of characteristic 2, with two properties: (1) The series represents a ...

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votes

**1**answer

490 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 ...

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439 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 ...

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**4**answers

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 ...

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votes

**1**answer

692 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} = ...

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votes

**1**answer

523 views

### Hankel determinants of symmetric functions

The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions.
...

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**1**answer

685 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 ...

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votes

**2**answers

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 ...

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**1**answer

672 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 ...

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**2**answers

682 views

### Is there a known formula for the number of SSYT of given shape with partition type?

Let $s_{\lambda}$ and $m_{\lambda}$ be the Schur and monomial symmetric functions indexed by an integer partition $\lambda$ ($\ell(\lambda)$ is the number of parts of $\lambda$ and $m_i(\lambda)$ is ...

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**2**answers

859 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 ...

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**1**answer

1k 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 ...