All Questions
Tagged with symmetric-groups linear-algebra
11 questions
12
votes
2
answers
1k
views
The character table of the symmetric group modulo m
Let $S_n$ be the symmetric group and $M_n$ the character table of $S_n$ as a matrix (in some order) for $n \geq 2$.
Question: Is it true that the rank of $M_n$ as a matrix modulo $m$ for $m \geq 2$ ...
- 26.5k
9
votes
2
answers
245
views
Matrix invariants for simultaneous conjugation by a finite subgroup of $\textrm{GL}_n$
Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ ...
- 1,114
8
votes
2
answers
1k
views
A basis for Schur functors
Suppose $V$ is a finite-dimensional vector space (over $\mathbb{C}$) and $\lambda$ is a partition of $n$ (not necessarily the dimension). Let $S^\lambda(V)=(V^{\otimes n})_\lambda$ be the $\lambda$'th ...
- 7,952
7
votes
3
answers
2k
views
Sarrus determinant rule: references, extensions
SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS
An undergraduate came to me with an identity for 4x4 determinants that is actually correct:
$\det(A)=h(A)+h(RA)+h(R^{2}A)$
where R cyclically ...
- 73
7
votes
2
answers
403
views
Decomposition of tensors into symmetry classes according to Schur functors
I am mainly looking for references on this subject, as I was unable to find any, at least any that answers what I am looking for to a satisfactory degree.
As it is well-known and extremely easy to ...
- 2,792
5
votes
1
answer
941
views
What is a concomitant (and other questions on D.E. Littlewood's "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" )?
I'm trying to understand the paper "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" by D.E. Littlewood (link), but I'm having trouble overcoming the language ...
- 10.7k
5
votes
1
answer
309
views
On a proof involving Young symmetrizers acting on tensor spaces
I hope this is not too elementary for this site, but I already asked something similar on MSE which has not received any attention whatsoever. I am extremely unfamiliar with the algebraic/...
- 2,792
4
votes
0
answers
98
views
Ref. request: Enumerating elements of Bruhat cells
Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then
$$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$
where we embed the ...
- 809
3
votes
0
answers
242
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 ...
- 5,898
2
votes
2
answers
132
views
Invertibility of one matrix constructed by order n subgroup of symmetric group
Let $S_n$ be the symmetric group on $n$ elements $\{ 1,2,\dotsc,n \}$ and $G$ be a subgroup of $S_n$ of order $n$. Denote the elements in $G$ by $\{ \sigma_1,\dotsc,\sigma_n \}$. Let the matrix $A=(\...
- 21
1
vote
0
answers
88
views
On the real and finite field rank of a $0/1$ matrix - II
Let $M\in\{-\ell,\dots,-1,0,+1,\dots,+\ell\}^{n\times n}$ be a matrix of rank $r$ where $\ell\geq1$ such that there is a permutation matrix in $\{0,1\}^{m\times m}$ of order $2\ell$.
Fix a permutation ...
- 13.9k