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6
votes
0answers
172 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
132 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
141 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
407 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
262 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
148 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?
3
votes
0answers
66 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 ...
3
votes
0answers
196 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 ...
2
votes
0answers
108 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
236 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
203 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
74 views

An inequality for elementary symmetric means of a periodic sequence

Given a tuple of positive numbers tuple $X = (x_1, x_2, \dots,)$, define the $k$th elementary symmetric mean of the first $n$ entries to be $$ S(X, n, k) := \frac{\displaystyle\sum_{i_1 < \cdots ...
1
vote
0answers
48 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
150 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
146 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) ...
0
votes
0answers
51 views

Reference request: two parameter analogy of the Waring formula.

Let $$p_k(t_1, \ldots, t_m)=t_1^k + \cdots + t_m^k$$ be the power sum symmetric functions and $$e_n(t_1, \ldots, t_m)=\sum_{1 \leq i_1 < \cdots < i_m\leq m} t_{i_1}\cdots t_{i_n}$$ the ...
0
votes
0answers
243 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 ...