# Questions tagged [symmetric-polynomials]

The tag has no usage guidance.

69 questions
Filter by
Sorted by
Tagged with
45 views

### Symmetric functions for multidimensional variables

I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize. For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$...
368 views

303 views

### Polynomial defined recursively by a resultant

Cross posting from MSE. Definition: For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...
175 views

### An explicit formula for characteristic polynomial of matrix tensor product [closed]

Consider two polynomials P and Q and their companion matrices. It seems that char polynomial of tensor product of said matrices would be a polynomial with roots that are all possible pairs product of ...
204 views

### Possible values of symmetric functions evaluated on quaternions

Let $i,j,k$ the units of quaternions, in particular $i^2=j^2=k^2=-1, ijk=-1$. We will use non commutative variables $x,y,z$. Define $sym_{a,b,c}$ to be the polynomial made of the sum of monomials ...
211 views

3k views

159 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 $i$ ...
147 views

I would like to construct (or determine the existence/inexistence) of a polynomial $p(x_1,...,x_k, y_1,...,y_n)$ satisfying the following properties: $p$ is symmetric with relation to the variables $... 2answers 657 views ### On a positivity property of Hall-Littlewood polynomials Here's the new, more thought through version. Consider a sequence of nonnegative integers$\lambda=(\lambda_1,\ldots,\lambda_n)$with$\lambda_i\ge \lambda_{i+1}+2$(the weight$\lambda-2\rho$is ... 0answers 144 views ### Behavior of elementary symmetric polynomials near zero sets It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points$\Lambda_{k}$in$x \in \mathbb{R}^{n}$with$\sigma_{k}(...
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 ...