All Questions
48 questions
54
votes
4
answers
5k
views
How many square roots can a non-identity element in a group have?
Let $G$ be a finite group. Let $r_2\colon G \to \mathbb{N}$ be the square-root counting function, assigning to each $g\in G$ the number of $x\in G$ with $x^2=g$. Perhaps surprisingly, $r_2$ does not ...
- 805
32
votes
3
answers
3k
views
Order of products of elements in symmetric groups
Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying
$1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$
whose product has order $...
- 19.6k
20
votes
1
answer
586
views
$q$-(and other)-analogs for counting index-$n$ subgroups in terms of Homs to $S_n$?
The following formula of astonishing beauty and power (imho):
$$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
- 24.9k
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
16
votes
2
answers
818
views
Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$
$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power $V^...
- 38.6k
16
votes
1
answer
1k
views
Tensor power of the natural representation of Sn
The symmetric group $S_n$ acts over $V=\mathbb{R}^n$ by permuting the canonical basis.
So it acts over $V^{\otimes p}$ with a diagonal action (acts the same over each element of the tensor product).
...
- 583
14
votes
4
answers
778
views
Largest permutation group without 2-cycles or 3-cycles
The largest permutation group without 2-cycles is $A_n$, which has size $n!/2$. I think the largest permutation group without 2-cycles or 3-cycles is much smaller, but I can't figure out if it should ...
- 143
14
votes
3
answers
660
views
Which partitions realise group algebras of finite groups?
Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$).
Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...
- 26.5k
14
votes
1
answer
388
views
Identity involving zonal polynomials and $\operatorname O(N)$ irrep dimensions
$\DeclareMathOperator\U{U}\DeclareMathOperator\O{O}$Schur functions $s_\lambda(x)$ with $\lambda\vdash n$ are simultaneously the irreducible characters of the unitary group $\U(N)$ and proportional to ...
- 2,552
13
votes
2
answers
1k
views
Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ?
I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...
- 24.9k
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
11
votes
1
answer
569
views
Counting symmetric subgroups of symmetric groups
This question is related to, but much more specific than, this one.
For $k \leq n$, let $a(k,n)$ denote the number of conjugacy classes of subgroups of the symmetric group $S_n$ which are isomorphic ...
- 2,753
10
votes
2
answers
598
views
Product of two reflections lying in a parabolic subgroup of a Coxeter group
Let $(W,S)$ be a Coxeter group, $I\subseteq S$ a subset of simple reflections, and $W_I \subseteq W$ the corresponding parabolic subgroup (we could also assume $|W_I|<\infty$, if needed).
Let also ...
- 165
10
votes
1
answer
199
views
Examples of differential towers of groups
For $r \in \mathbb{Z}_{>0}$, we say that a tower $1=G_0 \subseteq G_1 \subseteq \cdots$ of finite groups is an $r$-differential tower if for all $n$ the branching rules for restriction of ...
- 2,753
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
1
answer
683
views
Is $\operatorname{PSL}(2,q)$ the most quasirandom group?
Is the following statement true?
Every finite group $G$ has a non-trivial irreducible representation of dimension $O(\lvert G\rvert^{1/3})$.
Context: Groups with no small irreducible representations ...
- 986
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
2
answers
713
views
Total sum of squares of characters of the symmetric group $\mathfrak{S}_n$
In my earlier MO post, I proposed the double sum $\sum_{\mu\vdash n}\sum_{\lambda\vdash n}\chi_{\mu}^{\lambda}$ regarding characters of the symmetric group $\mathfrak{S}_n$. Soon after, I started ...
- 43.2k
7
votes
1
answer
326
views
Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$
Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...
7
votes
2
answers
1k
views
If you take a standard representation of a symmetric group, take an alternating tensor power of it, what groups appear as stabilizers of vectors?
I'm particularly interested in the case $\Lambda^3 \mathbb{F}_3^n$, and specifically, just stabilizers of vectors that satisfy the two conditions (i) there are no zero coordinates (in the basis ...
- 81
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
6
votes
1
answer
208
views
Sum identities with immanants
For $\chi$ being an irreducible character of the symmetric group $S_n$ and being $M$ a complex $n\times n$-matrix, I would like to show
$$
\sum_{\sigma, \rho \in S_n} \overline{\chi(\sigma)} \chi(...
- 93
5
votes
2
answers
292
views
Field with one element look at counting index-$n$ subgroups in terms of Homs to $S_n$, generalization to $F_{1^k}$?
Main idea shortly: As we discussed recently MO272045, there is beautiful fomula which
counts index-n subgroups in terms of homomorphisms to $S_n$.
Let me give "field with one element" interpretation ...
- 24.9k
5
votes
2
answers
390
views
Dimensions of Jordan blocks associated to representations
Given a linear representation $\rho$ of $SL_n(\mathbb C)$ of finite dimension $m$,
the image $\rho(U)$ of a maximal unipotent Jordan block $U\in SL_n$ decomposes
into generally several Jordan blocks ...
- 17.9k
5
votes
1
answer
264
views
Group not leaving subset invariant
Let $Y,X$ be two sets of size n,m. Let $Y\subset X$.
What is the maximal group(in size) $G< Sym(X)$ such that gY=Y imply that $g=1$?
Here I mean that the only permutation which permutes elements of ...
- 2,179
4
votes
1
answer
197
views
Moment integrals and determinants
Let $USp(2n)$ be the compact symplectic group of size $2n$, $dA$ its Haar measure
of total mass one, and $\det(1−A)$ being computed for the standard representation of
$A\in USp(2n)$ as a matrix of ...
- 43.2k
4
votes
2
answers
634
views
Bits and orbits
I know this title makes what I am about to ask sound like an off topic CS theory question but please bear with me because I assure you that it is not! (Well mostly, actually I am about ~90% certain ...
- 2,392
4
votes
1
answer
216
views
Representation of $\mathrm{AGL}(V)$ on the homology of the poset of affine subspaces of $V$
Let $V$ be an $n$-dimensional vector space over a finite field $F$ (of order $q$). Denote by $\mathrm{AGL}(V)$ the group of invertible affine transformations of $V$; so $\mathrm{AGL}(V)$ consists of ...
- 38.6k
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
2
answers
1k
views
numbering the squares of a rectangular grid, was: counting sequences of pairs
Hi,
Barry Cipra has rephrased my question in far superior clarity and brevity in an addendum to his answer below. I quote:
"If you number the squares of an $m×n$ grid, you can let three groups act ...
3
votes
2
answers
220
views
Factorizations of an $n$-cycle in $S_n$ into a product $xy$ where $|x| = 2, |y| = 3$
Let $S_n$ be the symmetric group on $n$ letters. When (and how) can an $n$-cycle in $S_n$ be factored into a product $xy$, where $x,y$ have orders 2,3 respectively?
More precisely, I'd like to ...
- 7,527
3
votes
1
answer
100
views
Exponent of unit group of Dynkin quiver algebras
Let kQ be the path algebra of a Dynkin quiver over a finite field with $q=p^n$ elements.
Let $f(Q,p^n)$ denote the exponent of the unit group of kQ. Is there an explicit formula for $f(Q,q)$?
- 26.5k
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
2
answers
533
views
elements in the weyl group
Let W be the Weyl a group of a semisimple simply connected group over C.
Let I={1,...,r} the set of simple roots.
For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
- 3,472
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
2
answers
557
views
Is there formula name and proof for this theorem ? [closed]
The formula answers: how many tuples $(\sigma_1,\sigma_2,...,\sigma_n)$ of elements of a given group G such that
(1) $\sigma_i\in C_i$ , where $C_i$ stands for conjugacy class.
(2) $\sigma_1\...
- 11
1
vote
1
answer
312
views
On the number of structure of $F_p[G]$-modules
Let $A$ be an abelian group and $G$ be a group. A short exact sequence of
groups like $1\longrightarrow A\longrightarrow E\longrightarrow
G\longrightarrow 1$ is called an extension. We say that $E$ is ...
- 463
1
vote
1
answer
511
views
Actions of $Z_n$ and actions of $Z_{n-1}$
I am playing with some questions concerning connections between
certain poset partitions and their linear extensions. This is not
my usual playground, I just happened to stumble upon something.
When ...
- 327
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
1
answer
280
views
Homotopy Equivalence of Posets for the Weyl Group
How do I go about proving Quillen's Homotopy Equivalence of Chevalley (subgroups of) automorphism groups of finite classical Lie algebras. Given $\sigma$ , such that Chevalley’s construction $GU$$n$($...
0
votes
1
answer
302
views
Lower bound of the largest irreducible character degree of alternating group $A_n$
$\newcommand\cd{\mathrm{cd}}$Let $A_m$ and $A_n$ be two alternating groups and $15\le m+2 \le n$. Denote $\cd_m$ and $\cd_n$ as the largest irreducible character degree of $A_m$ and $A_n$, ...
- 1
0
votes
1
answer
283
views
Is any abelian subgroup of a semidirect product isomorphic to a direct product of abelian subgroups? [closed]
Let $H$ and $K$ be groups and $V$ an abelian subgroup of the semidirect
product $\ H\rtimes K$. Do there exist abelian subgroups $H^{\prime }\leq H$
\ and $K^{\prime }\leq K$ \ such that $V\cong H^{\...
- 463
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
-1
votes
1
answer
438
views
Subgroup of the semidirect product of two subgroups with coprime orders [closed]
It is well known that if $\gcd (|H|,|K|)=1$ then all subgroups
of $H\times K$ are of the form $H^{\prime }\times K^{\prime }$ such that $H^{\prime}$ is a subgroup of $H$ and $K^{\prime}$ is a subgroup ...
- 463