# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...