All Questions
Tagged with co.combinatorics rt.representation-theory
82 questions
17
votes
0
answers
402
views
Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )
Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
- 24.9k
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
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):
$$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...
- 24.9k
19
votes
2
answers
1k
views
Explicit invariant of tensors nonvanishing on the diagonal
The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
- 148k
15
votes
2
answers
851
views
What are the periodic Dyck paths?
I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
- 26.5k
35
votes
4
answers
4k
views
How does this relationship between the Catalan numbers and SU(2) generalize?
This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote inspired by it, except that Math Overflow didn't exist then.
As ...
- 118k
103
votes
3
answers
6k
views
Why do combinatorial abstractions of geometric objects behave so well?
This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference.
Here are two examples of the kind of combinatorial abstractions of geometric ...
- 24.2k
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
16
votes
3
answers
2k
views
Integration of a function over 7-sphere
Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$.
The problem is finding or approximating the ...
- 163
6
votes
1
answer
407
views
hooks and contents: Part I
For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively.
R Stanley proved the following ...
- 43.2k
31
votes
6
answers
5k
views
What is known about this plethysm?
Let $S^{\lambda}$ be a Schur functor. Is there a known positive rule to compute the decomposition of $S^{\lambda}(\bigwedge^2 \mathbb{C}^n)$ into $GL_n(\mathbb{C})$ irreps?
In response to Vladimir's ...
- 156k
20
votes
3
answers
840
views
Is there an analogue of the hive model for Littlewood-Richardson coefficients of types $B$, $C$ and $D$?
If $V_\lambda$, $V_\mu$ and $V_\nu$ are irreducible representations of $\operatorname{GL}_n$, the Littlewood-Richardson coefficient $c_{\lambda\mu}^\nu$ denotes the multiplicity of $V_\nu$ in the ...
- 313
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
19
votes
2
answers
2k
views
Combinatorics problem related to Motzkin numbers with prize money I
Here a combinatorics problem. I offer 30 euro for a proof and 100 bounty points for a counterexample:
Let $n \geq 2$.
An $n$-Kupisch series is a list of $n$ numbers $c:=[c_1,c_2,...,c_n]$ with $c_n=1$...
- 26.5k
13
votes
6
answers
3k
views
A sum involving derivatives of Vandermonde
Consider the standard Vandermonde $V(x_1, \ldots, x_n) = \prod_{i < j} (x_i - x_j)$.
I am intersted in the calculation of the following expression for fixed $k$:
$$\sum_i (x_i)^k (d/dx_i)^k V(x_1 , ...
- 24.9k
11
votes
2
answers
558
views
Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
- 26.5k
11
votes
1
answer
569
views
Counting symmetric subgroups of symmetric groups
This question is related to, but much more specific than, this one.
For $k \leq n$, let $a(k,n)$ denote the number of conjugacy classes of subgroups of the symmetric group $S_n$ which are isomorphic ...
- 2,753
9
votes
1
answer
1k
views
A basic question about Young symmetrizers
This is probably elementary for experts on the representation theory of the symmetric group, but I did not find the answers I need by a cursory look at the usual textbooks (they could be there, but I ...
9
votes
1
answer
683
views
Is $\operatorname{PSL}(2,q)$ the most quasirandom group?
Is the following statement true?
Every finite group $G$ has a non-trivial irreducible representation of dimension $O(\lvert G\rvert^{1/3})$.
Context: Groups with no small irreducible representations ...
- 986
7
votes
1
answer
272
views
SYT and contents of a partition
Let $\lambda$ be an integer partition, denote the number of Standard Young Tableaux of shape $\lambda$ by $f_{\lambda}$. This number is computed by the formula
$$f_{\lambda}=\frac{n!}{\prod_{u\in\...
- 43.2k
6
votes
2
answers
309
views
Permanent of Nakayama algebras
See https://en.wikipedia.org/wiki/Nakayama_algebra for the definition of Nakayama algebras and define the permanent of such an algebra to be the permanent of its Cartan matrix.
(all algebras are ...
- 26.5k
5
votes
1
answer
771
views
Known decomposition of $\bigwedge^k Sym^d \mathbb C^n$ in special cases?
Let $V = \mathbb C^n$. Consider the plethysm $\bigwedge^k Sym^d V$ as a representation of $GL(V)$. In what special cases (e.g., for what $k$, $d$, and $n$) is this representation's decomposition into ...
- 326
4
votes
1
answer
700
views
Total sum of characters of the symmetric group $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.2k
3
votes
0
answers
203
views
What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?
Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
- 24.9k
37
votes
1
answer
1k
views
Errata for Fulton's "Young tableaux"
Fulton's Young tableaux is one of the best texts on the subject, one which I
often recommend and cite for reference. Unlike Fulton/Lang and
Fulton/Harris,
it is neither an early-dawn draft nor a ...
30
votes
1
answer
2k
views
Is there an accessible exposition of Gelfand-Tsetlin theory?
I'm hoping to start an undergraduate on a project that involves understanding a bit of Gelfand-Tsetlin theory, and have been tearing my hair out looking for a good reference for them to look at. ...
- 44.7k
29
votes
0
answers
1k
views
Linking formulas by Euler, Pólya, Nekrasov-Okounkov
Consider the formal product
$$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$
(a) If $z=2$ then on the one hand we get Euler's
$$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$
on the ...
- 43.2k
25
votes
7
answers
3k
views
Computer package for representation theory of the symmetric group
Is there a computer algebra package in which I can compute the following for representations of a specific symmetric group (e.g. $S_7$):
(1) $V \otimes W$
(2) $S_\lambda V$, where $S_\lambda$ is a ...
21
votes
14
answers
3k
views
Applications of Representation Theory in Combinatorics
What are the examples of interesting combinatorial identities (e.g. bijection between two sets of combinatorial objects) that can be proved using representation theory, or has some representation ...
21
votes
1
answer
1k
views
Bounding Schur symmetric polynomials on the unit circle
Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by
\begin{equation}
s_\lambda(x_1,\ldots, x_n) = a_{\lambda + \delta}(x_1, \...
- 4,466
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
0
answers
824
views
Capelli determinant = Duflo ( determinant) - was it known ?
Question briefly. Was this fact known: Capelli determinant = Duflo (determinant) ? (This is an equality of the two central elements in universal enveloping of Lie algebra $gl_n$).
I googled a lot ...
- 24.9k
15
votes
2
answers
1k
views
Branching rule from symmetric group $S_{2n}$ to hyperoctahedral group $H_n$
Embed the hyperoctahedral group $H_n$ into the symmetric group $S_{2n}$ as the centralizer of the involution $(1, 2) (3, 4) \cdots (2n-1, 2n)$ (cycle notation). Label representations of $S_{2n}$ by ...
- 10.7k
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
4
answers
1k
views
actions of the hyperoctahedral group
I am looking for actions (i.e., permutation representations) of the hyperoctahedral group $H_n$ (also known as the group of signed permutations) studied in the literature, i.e., homomorphisms from $...
- 5,822
13
votes
2
answers
862
views
Basis of coinvariant algebra on which reflection group acts as regular representation
This question is almost a duplicate of a question of Christian Stump, except that Christian seems to ask about an isomorphism to irreducible representations rather than the regular representation: ...
- 24.2k
13
votes
1
answer
991
views
Why do we care about Schur Positivity
Some of the most important open problems in Algebraic Combinatorics concern the Schur positivity of classes of symmetric functions. Why is this an important property to have?
13
votes
1
answer
637
views
trace and involution permutations: Part I
Let $\operatorname{Inv}(\mathfrak{S}_n):=\{\pi\in\mathfrak{S}_n: \pi^2=1\}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$. Denote $I_n:=\#\operatorname{Inv}(\mathfrak{S}_n)$. Let $\...
- 43.2k
12
votes
2
answers
587
views
Bounding weight multiplicities by number of certain Coxeter elements
This question concerns lower bounds of certain weight multiplicities in finite dimensional representations of algebraic groups (or Lie groups, Lie algebras).
Let's say $G$ is a simple algebraic group ...
- 313
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
12
votes
1
answer
1k
views
Why is this character expression an integer?
Let $\gamma$ be an $n$-dimensional complex representation of a finite group $G$ with character $\chi$ and let $e=c_0, c_1, ..., c_{\ell}$ be a set of conjugacy class representatives for $G$. In the ...
- 2,753
11
votes
2
answers
515
views
Using irreducible characters of the orthogonal group as basis for homogeneous symmetric polynomials
The irreducible characters of the orthogonal group $O(2N)$ are given by
$$ o_\lambda(x_1,x_1^{-1},...x_N,x_N^{-1})=\frac{\det(x_j^{N+\lambda_i-i}+x_j^{-(N+\lambda_i-i)})}{\det(x_j^{N-i}+x_j^{-(N-i)})}...
- 1,549
11
votes
2
answers
916
views
Young-Fibonacci tableaux, content, and the Okada algebra
Using the French convention, the content of the $i \times j$ box in the Young diagram of a partition $\lambda$ is $i-j$. Now if
$\lambda$ is partition of $n$ and $\sigma_\lambda: S_n \longrightarrow ...
- 2,137
10
votes
1
answer
627
views
Littlewood-Richardson coefficients for Jack symmetric functions
Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(\alpha)$.
We define a scalar product $\langle \cdot,\cdot\rangle_\alpha$ on $\Lambda$ by setting $\langle p_\mu,p_\nu\rangle_\...
- 158
10
votes
1
answer
400
views
Derived equivalences of Dyck paths
Call two Dyck paths $D_1$ and $D_2$ derived equivalent in case their corresponding Nakayama algebras are derived equivalent (The Dyck path of a Nakayama algebra with a linear quiver is just the top ...
- 26.5k
10
votes
1
answer
3k
views
An enumeration problem for Dyck paths from homological algebra
In their article "On n-Gorenstein rings and Auslander rings of low injective dimension" Fuller and Iwanaga gave a homological characterisation of 2-Gorenstein Nakayama algebras with global ...
- 26.5k
10
votes
4
answers
939
views
Is there a non-explicit characterization of the Specht modules?
It is a basic fact about the symmetric group $S_n$ that its irreducible representations are indexed by partitions of $n$.
My question is, can the association between partitions and irreps be ...
- 258
9
votes
1
answer
878
views
Littlewood Richardson rule and seminormal basis of Specht modules
Background
Seminormal Basis of Specht modules of $\mathfrak{S}_n$
Let $\lambda$ be a partition of $n$. A $\lambda$-tableau is a
bijection $\mathfrak{t}:\lambda \to \{1,2,...,n\}$. We say a ...
- 1,413
9
votes
1
answer
384
views
Smith Normal Form of a Cayley Graph of the Symmetric Group
Let $A_n$ be the adjacency matrix of the Cayley graph $\text{Cay}(S_n,C_n)$ where $C_n \subseteq S_n$ is the conjugacy class of $n$-cycles of the symmetric group $S_n$. Since the generating set of ...
- 525
9
votes
0
answers
398
views
When do almost all these invariants of tensors vanish?
Let $A,B,C,D$ be $n$-dimensional vector spaces over a field $k$.
There is a natural homomorphism from the $mn^m$th tensor power $A^{\otimes (m n^m)} $ of $A$ to $k$ given by the determinant map $A^{\...
- 148k