All Questions
14 questions with no upvoted or accepted answers
20
votes
0
answers
451
views
Row of the character table of symmetric group with most negative entries
The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...
- 24.2k
12
votes
0
answers
513
views
Converse of Frobenius
Enumerate the elements of a finite group $G$ as follows: $g_1,g_2,\dots,g_n$. Introduce $n$ variables indexed by the elements of $G$: $x_{g_1},\dots,x_{g_n}$.
Consider the matrix $X_G$ with entries $...
- 43.2k
10
votes
0
answers
436
views
Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?
$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
...
- 24.9k
9
votes
0
answers
275
views
pattern-avoiding permutations vs multi-core partitions
Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Given the pattern $\sigma=k(k-1)\cdots321$, let $I_n(\sigma)$ be the number of involutions in $\mathfrak{S}_n$ that avoid the pattern $\sigma$. ...
- 43.2k
8
votes
0
answers
247
views
Computing the number of elementary abelian p-subgroups of rank 2 in $GL_{n}(\mathbb{F}_{p})$
Let $p$ be a prime number, and let $\mathbb{F}_{p}$ be a finite
field of order $p$. Let $GL_{n}(\mathbb{F}_{p})$ denote the general linear
group and $U_{n}$ denote the unitriangular group of $n\times ...
- 463
7
votes
0
answers
239
views
Combinatorial Avatar of Irrep Dimensions Dividing the Order of the Group
Suppose $G$ is a finite group and $V$ a complex irreducible representation. Let $v\in V$ be a sufficiently generic vector, and consider its orbit, $O(v)=\{gv|g\in G\}$. As a naive attempt to ...
- 85.6k
4
votes
0
answers
531
views
How many non-isomorphic groups share the same character table?
I have been thinking for a while about how to enumerate all finite groups. The classical way e.g. here would be to go via latin squares and then try to calculate how many of those obey associativity. ...
4
votes
0
answers
153
views
partial sum over characters of symmetric group
It is well known that irreducible characters of the symmetric group satisfy
orthogonality relations,
$$ \sum_{\mu \in P(n)} \chi_\mu^\lambda\chi_\mu^\omega=z_\lambda\delta_{\lambda,\omega},\quad
\...
- 2,552
3
votes
0
answers
210
views
How many conjugacy classes of elementary abelian subgroups of order $p^2$ does $\operatorname{GL}_{4}(\Bbb Z / p\Bbb Z)$ have?
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Let $f\in \Hom((\Bbb Z/p\Bbb Z)^2,\GL_{4}(\Bbb Z / p\Bbb Z))$ be an injective homomorphism. What is the number of ...
- 463
2
votes
0
answers
77
views
roots and embeddings
Let $G$ be a connected reductive group over an algebraically closed field, can we always find an embedding let $\rho:G\rightarrow GL_n$, that sends a Borel pair $(B_G,T_G)$ to $(B,T)$ and center to ...
- 3,472
2
votes
0
answers
255
views
Summing Characters of the Symmetric Group over Derangements (Enumerative Combinatorics: Vol. II Ex. 7.63)
The following exercise is out of Stanley's Enumerative Combinatorics: Vol. II (Ex. 7.63):
For $\lambda \vdash n$ define $d_\lambda = \sum_{w \in D_n} \chi^\lambda(w)$ where $D_n$ is the set of all ...
- 525
1
vote
0
answers
130
views
Relationship between the symmetric group representation (Specht module) of a Young diagram and the Young diagram obtained by deleting one row
Suppose $\lambda$ is a Young diagram, and $\lambda'$ is obtained by deleting one particular row of $\lambda$. Is there any relationship between the symmetric group representation (Specht module) ...
- 25
0
votes
0
answers
95
views
Class multiplication coefficients of symmetric groups
My question is that I was working with some counting problems, and finally the answer should be
$$
\nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\...
0
votes
0
answers
116
views
Multivariate polynomial representations of the infinite dihedral group
The presentation given in Wikipedia for the infinite dihedral group is
$$\langle r,s\mid s^2 =1, srs = r^{-1}\rangle.$$
Let $[R]$ denote the infinite set of reciprocal partition polynomials $R_n(u_1,...
- 10.5k