All Questions
Tagged with enumerative-combinatorics rt.representation-theory
20 questions
1
vote
0
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139
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Integral convex polytopes formed from the weight diagrams of representations of $\mathfrak {sl}_4$($\mathbb{C}$)
I'm a student studying undergraduate abstract algebra and doing a summer research project in the mathematics department at my school. I'm barely familiar with the rudiments of representation theory; I ...
- 111
6
votes
0
answers
365
views
Is this just a numerical accident or what?
In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation
$$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m}
=\prod_{...
- 43.1k
4
votes
0
answers
163
views
An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
- 43.1k
2
votes
0
answers
87
views
Reference request on Plancherel measure for partitions whose parts differing by more than $1$
Given an (unrestricted) integer partition $\lambda$ of $n$, let $f_{\lambda}$ denote the number of standard Young tableaux (SYT) of shape the Young diagram $Y(\lambda)$ of $\lambda$. Then,
$$\sum_{\...
- 43.1k
7
votes
0
answers
152
views
Inequality of product of discrete cosines
Let $k,a,b,c$ be odd positive integers. Consider the following inequality:
$$
\sum_{x,y \in [k]} \cos^a\bigg(\frac{2\pi}{k}\cdot x\bigg) \cdot \cos^b\bigg(\frac{2\pi}{k}\cdot y\bigg) \cdot \cos^c\bigg(...
- 71
5
votes
1
answer
804
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Expressing symmetric function in power-sum basis
I am trying to prove the following identity
\begin{equation}
\prod_{i=1}^{m}(1-x_{i}z)^{-u}\prod_{j=1}^{n}(1-y_{i}z)^{-v} \prod_{i=1}^{m}\prod_{j=1}^{n}(1-(x_i +y_j)z)^{-w}\\ = \sum_{\lambda, \mu}c_{\...
- 685
0
votes
0
answers
171
views
Total sum of characters over partitions with distinct parts
In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...
- 43.1k
7
votes
2
answers
712
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 ...
- 43.1k
4
votes
1
answer
698
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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
$$\sum_{\lambda\vdash n}\...
- 43.1k
6
votes
2
answers
612
views
Counting $\pm 1$ and $0$'s in the character tables of $\frak{S}_n$
Let $\chi_{\mu}^{\lambda}$ denote a value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu, \lambda\vdash n$. When $\mu=(n)$, then it's known that
$$\sum_{\lambda\vdash n}\...
- 43.1k
1
vote
0
answers
90
views
Dimension of a certain space of symmetric functions: Part II
This is the second installment of my earlier MO question.
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. Denote the set of all partitions with distinct parts by $\...
- 43.1k
4
votes
0
answers
205
views
Dimension of a certain space of symmetric functions: Part I
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$.
QUESTION. Consider the ...
- 43.1k
5
votes
0
answers
131
views
Identity for classes of plane partitions
There are several classes of plane partitions in the literature.
Among these, let's look at the enumeration of three of them: the symmetric (SPP), totally symmetric (TSPP) and totally symmetric and ...
- 43.1k
15
votes
1
answer
749
views
Character theoretic proof of the Littlewood–Richardson rule?
The Littlewood–Richardson coefficients are the multiplicities
$$
c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu))
$$
and the Littlewood–Richardson rule says that ...
- 1,413
16
votes
1
answer
804
views
Existence of a faithful irreducible representation using Möbius function
Let $G$ be a finite group, $L(G)$ its subgroup lattice and $\mu$ the Möbius function.
Consider the Euler totient of $G$ defined as follows:
$$ \varphi(G) = \sum_{H \le G}\mu(H,G) |H| $$
Let $X=\{M_1, \...
7
votes
1
answer
366
views
a new representation for Eulerian numbers?
The Eulerian numbers enjoy many different presentations among which I write the two-variable recursive definition: $A(n,0)=1$ and $A(n,k)=0$ for $k<0$ so that
$$A(n,k)=(k+1)A(n-1,k)+(n-k)A(n-1,k-1)....
- 43.1k
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 $\sigma$. ...
- 43.1k
5
votes
1
answer
151
views
Counting the orbits of a set of tabloids under the action of a Young subgroup
Let $\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k)$ and $\mu = (\mu_1 \geq \mu_2 \geq \cdots \geq \mu_\ell)$ be partitions of a positive integer $n$. As in Fulton's book on Young ...
1
vote
1
answer
220
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Reference request: $\chi^{\lambda'}(\sigma) = (-1)^{n-\ell(\sigma)} \chi^\lambda(\sigma),$ for characters of the symmetric group
I'm looking for a text I could cite that explicitly states the following result: for $\chi^\lambda$ the irreducible character of the symmetric group indexed by the partition $\lambda$, and for $\sigma ...
- 2,151
3
votes
0
answers
113
views
Enumerating simple algebraic groups and their irreducible representations
Motivation
Everything is over an algebraically closed field.
Given a faithful representation $G \to \textrm{GL}(V)$, one may try to pin down what the group $G$ exactly is (e.g., the in the case of ...
- 5,484