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This problem has little to do with representation theory, other than the fact that the eigenvalues of a matrix of finite order are roots of unity. If $\alpha$ lies in a cyclotomic extension $K$ of $\m …

I will confine my comment to ordinary characters. I suspect that if $p$ is not too close to $n$ then the largest character value will be roughly of size $\sqrt{(n-p)!}$. Let $\mu$ be a partition of $ …

For convenience consider the representation $Y=V_n\oplus V_{n-1,1}$ instead of $V_{n-1,1}$. Then the multiplicity of the representation of $S_n$ indexed by the partition $\lambda$ of $n$ in the $k$th …

In the language of symmetric functions, you are computing the plethysm $e_j[e_k]$ (also denoted $e_k\circ e_j$) of elementary symmetric functions. In terms of $\mathrm{GL}(n,\mathbb{C})$ representatio …

In general, if $(\lambda,\mu)$ is a bipartition of $n$, then $$ \prod_i(p_{\lambda_i}(x)+p_{\lambda_i}(y))\cdot\prod_j (p_{\mu_j}(x)-p_{\mu_j}(y)) = \sum_{(\alpha,\beta)} \chi^{\al …

The determinant of an $n\times n$ matrix with a subdiagonal of 1's, and 0's below the 1's, can be written as a sum over subsets of $S$ and interpreted in terms of Inclusion-Exclusion. See Enumerative …

The number of ways to place the balls so that the $j$th box holds exactly $\lambda_j$ balls is the number $N_{\lambda\mu}$ of matrices of nonnegative integers with row sum vector $\lambda$ and column …

The multiplicity of the irreducible character of $S_n$ indexed by the partition $\lambda$ of $n$ in the action of $S_n$ on itself by conjugation is the coefficient of the Schur function $s_\lambda$ i …

I gave an answer for inducing the trivial representation in a comment at Decomposing the conjugacy representation of Sym$(n)$ for small $n$. It is in terms of a plethysm that seems to be just as intra …

It can't be true in general that $m_{\mu,I}$ is always 0 or 1. That is because $K_{\mu,1^n}=f^\mu$, the number of standard Young tableaux of shape $\mu$. The maximum value of $f^\mu$ for $\mu\vdash n$ …

For type A, a combinatorial interpretation of the entries of the inverse matrix was given by O. Egecioglu and J.B. Remmel, A combinatorial interpretation of the inverse Kostka matrix, Linear Multiline …

The answer by Ottem shows that there is some way to write the generating function to correspond to a chain of syzygies. However, it is not true that every way works. For instance, let $G$ be the group …

The cohomology ring of the Grassmannian, an avatar known to Philip Hall.

It seems to me that the answer is no. For instance, let $\rho(G)$ consist of the $2\times 2$ signed permutation matrices, a group of order 8. Let $w=(2,1)$. The lattice $L$ is all of $\mathbb{Z}^2$ si …

Using standard symmetric function notation, we have \begin{eqnarray*} \sum_{n\geq 0}\sum_{\lambda,\mu\vdash n} \frac{1}{n!}\left(\sum_{\pi\in D_n}\chi_\lambda(\pi)\chi_\mu(\pi)\right) s_\l …