# Tagged Questions

The tag has no usage guidance.

1k views

176 views

### An identity related to partitions into $n$ parts and Schur polynomials

While working with Schur polynomials I found what seems like a nice identity, and I wonder if it has a simple proof. Notation: Suppose $d,n\in\mathbb{N}$, and $\lambda =(\lambda_1,\dots,\lambda_n)$ ...
190 views

### Most computationally efficient Littlewood-Richardson rule

There are many, many different versions of the Littlewood-Richardson rule: the original characterization via Yamanouchi words, Remmel's version, a description via the Poirier-Reutenauer bialgebra, the ...
246 views

### Schur polynomial, change of variable

Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$. Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ ...
1k 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 ...
737 views

### What is the most general “two in one row for A & in one column for B” theorem?

Let $A$ and $B$ be two Young tableaux, i. e. Young diagrams filled with the numbers $1$, $2$, ..., $n$ for some $n$ (not necessarily the same $n$). (They need not be semistandard.) (a) (Etingof's ...
543 views

### Cut-and-join equation and Schur function identity

This is somewhat related to my last MO post: sum of the character of the symmetric group Let $p_n$ be the $n$-th Newton symmetric function, and $s_{\nu}$ be the Schur function indexed by the ...
225 views

### Staircase Schur functions squared

Let $\Delta_n$ be the staircase-shaped partition $(n-1,n-2,\dots,1)$. Are there any non-obvious combinatorial objects that index $s_{\Delta_n}^2$? Here, $s_\lambda$ is the Schur function indexed by ...
759 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: sλ...
232 views

### Dickson/determinant type polynomial (updated)

For $2\leq \ell \leq k$, consider the polynomial $$P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k]$$ ...
143 views

225 views

79 views

### Bounding Schur polynomials of a particular shape

Consider Schur polynomials $s_\lambda$ with $\lambda = (2m, m, m, \ldots, m, 0)$ and $\ell(\lambda) = n$ (that is, $\lambda$ has $n$ rows). Here $m \gg n$, which, for the sake of concreteness, let's ...
102 views

### what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions.

I am trying to understand first how one can define the plethysm say $s_\lambda \circ s_\mu$ as a module in the regular representation of the symmetric group. 1)How is it connected to the plethysms ...
343 views

### principal specialization of projective Schur functions

Is there a nice/known formula for the (some) principal specialization of a projective (P or Q) Schur polynomial? That is, I am talking about a projective analogue of Stanley's hook content formula ...