The tag has no wiki summary.

learn more… | top users | synonyms

16
votes
0answers
245 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 ...
14
votes
0answers
200 views

An algebraic strengthening of the Saturation Conjecture

The Saturation Conjecture (proved by Knutson-Tao) asserts that $c_{n\mu,n\nu}^{n\lambda}\neq 0\Rightarrow c_{\mu,\nu}^{\lambda} \neq 0$, where $c$ denotes a Littlewood-Richardson coefficient and $n$ ...
8
votes
0answers
170 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
0answers
208 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
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
0answers
60 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
195 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
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?
4
votes
0answers
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 ...
3
votes
0answers
85 views

A “nice” Orthogonal Basis for Translation Invariant Symmetric Polynomials

It is going to be a rather long question, so I will first state it and then try to explain and motivate it. Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
3
votes
0answers
92 views

Efficiently computing (plethysm-like?)substitutions of symmetric functions

This is a rather technical question, it arose in connection of some calculations that I need to have better grasp of the question Formal group law over $\mathbb{F}_p$ and my own older one What is ...
3
votes
0answers
337 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
0answers
135 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 ...
3
votes
0answers
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 ...
2
votes
0answers
81 views

Permutation-invariant matrix representation

The question guide says that Mathoverflow is for research level mathematics. While I do not perform research in mathematics (I study quantum chemistry), I believe this question is research-level ...
2
votes
0answers
150 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 ...
2
votes
0answers
172 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) ...
2
votes
0answers
298 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 ...
2
votes
0answers
211 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 ...
1
vote
0answers
37 views

integral schur function over standard simplex

Let $T^d$ be the standard simplex, $$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{R}^{d}\mid\sum_{i = 1}^{d}{t_i} = 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\right\} $$ For any partition ...
1
vote
0answers
92 views

What is the definition of plethysm in the representation theory of permutation groups

Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams. I have seen the definition of plethysms in symmetric functions. I would like to understand the ...
1
vote
0answers
92 views

Evaluation of Macdonald Polynomials at $x_1=x_2=…=x_n = h$

This question is related to my previous question (here). Let $P_\lambda$(q,t) be the Macdonald polynomials with partition $\lambda$. Let $\Lambda$ denote the ring of symmetric functions over the ...
1
vote
0answers
80 views

Character sums over a fixed subset of skew tableaux

Let $f(\lambda)$ count the number standard young tableaux of shape $\lambda\vdash n$ and $\lambda=(\lambda_1,\cdots,\lambda_r)$. Let $\mu \vdash k$ be a partition for $k<n$. It is a consequence of ...
1
vote
0answers
124 views

counting how many boxes from a given Young tableau contribute to hook length made out of two YTs

Think of a Young diagram as a collection of rows with numbers of elements $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...
1
vote
0answers
25 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 ...
1
vote
0answers
66 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 ...
1
vote
0answers
84 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
0answers
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 ...
1
vote
0answers
288 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 ...
0
votes
0answers
61 views

how should this multinomial identity be written?

question if it is correct, how is the identity tagged (4) below usually written, and can the use of conjugate partitions be avoided? motivation i apologize for the length of this question - it's as ...