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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 ...
9 votes
0 answers
233 views

Hives for other root systems? [duplicate]

Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ have numerous combinatorial interpretations, including the hive model by Knutson and Tao, see here. For other root systems, there are also ...
Igor Pak's user avatar
  • 17k
8 votes
0 answers
240 views

Scalar products on symmetric functions behaving like the Macdonald scalar product

The Macdonald symmetric functions (or Macdonald polynomials) $P_\lambda(x)$ are orthogonal with respect to the Macdonald scalar product $$ \langle p_\lambda,p_\mu\rangle = \delta_{\lambda\mu}z_\...
Richard Stanley's user avatar
5 votes
1 answer
804 views

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_{\...
GGT's user avatar
  • 685
8 votes
1 answer
272 views

Branching rule for Specht modules over Kazhdan-Lusztig basis

Suppose $\lambda\vdash n$ is a partition and $S^\lambda$ is the associated irreducible representation of $S_n$. As we know from the branching rule, we have an isomorphism of $S_{n-1}$ modules $$S^\...
GossipM's user avatar
  • 221
5 votes
0 answers
103 views

Form on symmetric functions and their q,t- analogues

[Notations are as in Macdonald's Symmetric Functions and Hall Polynomials] The space of symmetric functions $\Lambda_{\mathbb{Q}}$ has a bilinear form defined by $ (p_\lambda, p_\mu)= z_\lambda \...
ArB's user avatar
  • 820
6 votes
0 answers
188 views

Word/Loop in $L(\Lambda)$

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, with Chevalley generators $e_i,f_i$ ($i=1,...,n$). Let $L(\Lambda)$ denote the irreducible module with highest weight $\Lambda$. Let $v$ denote ...
ArB's user avatar
  • 820
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 ...
Nathan Lindzey's user avatar
6 votes
0 answers
1k views

Frobenius formula

I know two formulas by the name of Frobenius. The first one computes the number $$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$ where $...
Gabriel's user avatar
  • 711
5 votes
0 answers
267 views

Tabloid Construction of Permutation Representation of Hyperoctahedral Group

For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\...
Max Hopkins's user avatar
9 votes
1 answer
650 views

A stronger version of a problem of Kenneth Brown using representations

Let $G$ be a finite group and $\mathcal{L}(G)$ its subgroup lattice. Let $\mu$ be the Möbius function on $\mathcal{L}(G)$. The reduced Euler characteristic of the order complex of the coset poset $\{ ...
Sebastien Palcoux's user avatar
2 votes
1 answer
381 views

Counting cosets in the Quotient of Weyl groups

Let $\Sigma=G/P$ be a flag variety of type $A$,i.e. $G=SL_{n+1}$ and is a stabilizer of the partial flag $0 \subset V_{d_1}\subset \cdots\subset V_{d_r}\subset V_{n+1}$ of length $r$. Let $W$ be its ...
MIQ's user avatar
  • 83
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 ...
4 votes
1 answer
246 views

Is there a nice form for the Frobenius characteristic of a border shape character?

Let $\chi^V$ be the character of a border strip Specht module, i.e. a Specht module for a skew tableau that contains no $2 \times 2$ square. I know that the Frobenius characteristic of $\chi^V$ is ...
user162496's user avatar
1 vote
1 answer
311 views

Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory

Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let $(n_0,\...
John Murray's user avatar
  • 1,090
2 votes
1 answer
352 views

Question about a proof in Graham and Lehrer's "Cellular algebras"

I'm sorry if this question is too basic for MO. I'm reading a paper by Graham and Lehrer "Cellular algebras" and have trouble understanding one step in a proof of a crucial theorem. I suppose that the ...
Balerion_the_black's user avatar
21 votes
1 answer
2k views

Conjectural identities for Young symmetrizers and Young-Jucys-Murphy elements

The following questions I have found in my own notes from about 3 years ago. Unfortunately, I lost much of the context; I believe I made these conjectures reading Okounkov-Vershik, arXiv:0503040v3, ...
darij grinberg's user avatar
6 votes
2 answers
881 views

What is the most general "two in one row for A & in one column for B" theorem?

Let $A$ and $B$ be two Young tableaux, i. e. Young diagrams filled with the numbers $1$, $2$, ..., $n$ for some $n$ (not necessarily the same $n$). (They need not be semistandard.) (a) (Etingof's ...
darij grinberg's user avatar