1
vote
0answers
33 views

Dimension of the space of tensors obtained by making partial symmetrizations and skew-symmetrizations.

Let $A=(a_{i_1\dots i_k})_{i_1,\dots,i_k=1}^n$ be a higher order cubic tensor or hypermatrix. The following two facts are well-known and are easy to prove: ${(\bf 1) }$ The dimension of the ...
1
vote
1answer
50 views

Derivation and application of Newton's identity

How is the following identity derived? $$\sum_{\ell =0}^{n-1}(-1)^\ell e_\ell s_{n-\ell}+(-1)^nne_n=0$$ Is there an example demonstrating the context in which this might be applied?
0
votes
0answers
30 views

How to compute the dimesnion of the image of the tensor product of Young symmetrizers?

The following identity is contained in J. Landsberg, Tensors: Geometry and Applications, Graduate Studies in Mathematics, v.128. (p.152): $$ S^3(A\otimes B\otimes C)= S_3S_{3}S_3\oplus ...
14
votes
1answer
165 views

Involutions, RSK and Young Tableaux

Let $S_n$ be the symmetric group on $n$ elements. The Robinson-Schensted-Knuth (RSK) correspondence sends a permutation $\pi\in S_n$ to a pair of Standard Young Tableaux $(P,Q)$ with equal shapes ...
1
vote
0answers
26 views

Characterisations of the RSK correspondence

I know of the following three definitions of the RSK correspondence: (i) Row insertion (or more generally, plactic insertion) (ii) Viennot's construction (iii) Fomin's growth diagrams However, all ...
1
vote
0answers
45 views

Proof of dimension formula for sl(3,C) reps

this is my first question on the forums, so forgive me if I did something wrong. I have tried to find an answer in here as well as on the usual places on the internet and was unsuccessful. So here ...
3
votes
1answer
63 views

Trying to understanding the proof of the fact that Kazhdan property (T) implies expanders.

I am trying to trying to understanding the proof of the fact that Kazhdan property (T) implies expanders. This is a result of Grigory Margulis. It is stated in Proposition 3.3.1 on Page 30 of the book ...
5
votes
1answer
68 views

Schur functors as spaces of “flag tensors”?

Consider the following construction: for a vector space $V$, define $W \subseteq \bigwedge^2 V \otimes V$ by $W = \langle\ \alpha \otimes v : v \in \text{Span}(\alpha) \ \rangle$, that is, $W$ is ...
1
vote
0answers
18 views

Finding an orthonormal basis for a gl(3) module

I'm trying to find an orthonormal basis for gl(3)-module V(ε1-ε3), where ε1-ε3 is the weight (1,0,-1) of the highest-weight vector. Using Gelfand-Tsetlin (/Zetlin/Zeitlin) patterns, I'm at the point ...
3
votes
1answer
53 views

Number of Cocharge Tableaux Summing to Fixed Numbers

First, some background: I will assume the anglophone conventions for Young tableaux in what follows. Given a standard Young tableau $T$ of shape $\lambda$, we can define the cocharge tableau $C(T)$ as ...
3
votes
1answer
159 views

Definition of the Young Symmetrizer

I have a question regarding the equivalence of two definitions of the young symmetrizer. First, some notation: let $\lambda$ be a partition of $n$. Given a $\lambda$-tableaux $T$, we define the row ...
1
vote
0answers
68 views

Number of Standard Young Tableaux of n cells

I know there is a 1-1 correspondance between the number of standard young tableaux of $n$ cells and the number of involutions in $S_n$. Number of involutions in $S_n$ satisfies the recurrence relation ...
1
vote
1answer
104 views

Number of Inequivalent Difference Sets In Elementary Abelian 2-groups

I have reason to believe that there is only one$(2^{2s+2},2^{2s+1}-2^s,2^{2s}-2^s)$- difference set (based on experimentation in GAP), up to equivalence/complementation, in any elementary 2-group of ...
8
votes
1answer
145 views

Theorem 1 chapter 8 of Fulton's Young Tableaux

I am reading Theorem 1 on page 110 of Fulton's Young Tableaux and have several questions on it. Let $E$ be a free module on $e_1,\ldots,e_m$ (for our purposes $E$ being a finite dimensional complex ...
0
votes
0answers
62 views

Counting Young tableaux

Let's say we have some shape $\lambda$ and we want to fill this shape with numbers $\{1, .., m\}$ in non-decreasing order in rows and columns. How many such numberings do we have? I can not find ...
0
votes
0answers
51 views

writting a code for finding the Kostant partition function

How to write a code in sage for finding the Kostant partition function for the elements of root lattice of rank 1 affine lie algebra $A_{1}^{(1)}$ which is defined as follows: $K(\beta)$ = the ...
3
votes
0answers
83 views

On Applications of the Murnagham Nakayama rule

The question is located below. In short, I am looking for an accessible explanation of the Murnagham Nakayama rule in relation to the following problem. Pardon the long setup. Let $Y$ be a standard ...
6
votes
0answers
147 views

Dimension of the space of algebraic Riemann curvature tensors

Given $n\in \mathbb N$, consider the vector space $\mathbb R^{n^4}$ whose elements I will denote by $(R_{abcd})$ with indices $a,b,c,d \in \{1, \dots, n\}$. This vector space is $n^4$-dimensional. The ...
3
votes
1answer
78 views

Bounds on Young Tableau Element locations

I'm having trouble finding some elementary results on the following. Let $Y$ be a standard Young Tableau of shape $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_n)$ with $N:=\sum_{i=1}^n\lambda_i$. My ...
3
votes
1answer
45 views

How to compute the weights of $\Gamma_{3,1}$ the irrep of $\mathfrak{sl}_3\Bbb C$

I am wondering about a combinatorial formula for computing the weights of the irreducible representations $\Gamma_{a,b}$ of $\mathfrak{sl}_3\Bbb C$. By $\Gamma_{a,b}$ I mean the irrep that has highest ...
4
votes
1answer
131 views

Formula for evaluation of character on a transposition

Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of ...
1
vote
0answers
130 views

Young tableaux: evaluate action of permutation

Consider the irreducible representation $V$ in the symmetric group $S_5$ corresponding to the Young diagram (these are meant to be boxes): $$[\;\;][\;\;] \\ [\;\;][\;\;] \\ [\;\;]\;\;\;\;$$ (a) List ...
6
votes
0answers
238 views

Young Tableaux as Matrices

These questions are motivated only by curiosity. Take a Young tableau of shape $(\lambda_1,\lambda_2,\ldots,\lambda_n)$, where $\lambda_1\geq\lambda_2\geq\ldots\geq \lambda_n)$. Is there any physical ...
5
votes
1answer
86 views

$S_n$ has only four (irred.) representations with degree $<n$ (for $n>6$)

I'm working on the following exercise: For $n\ge 7$, $S_n$ has no irreducible representations of dimension $m$ with $2\le m\le n-2$. There is a solution here but I'd like to follow the ...
8
votes
1answer
135 views

Fulton and Harris A.23

I am reading the appendix of Fulton and Harris pg. 459 and am trying to understand the following setup. Suppose $\lambda : \lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_k \geq 0$ is a partition of ...
3
votes
1answer
90 views

Stuck in proof of combinatorial identity - Fulton and Harris A.39

I'm trying exercise $A.39$ in Fulton and Harris. They suggest to first prove the formula $$|x_j^{l_i}| \prod_{j=1}^k(1-x_j)^{-1} = \sum |x_j^{m_i}| \hspace{1in} (\ast)$$ where the sum on the right ...
4
votes
2answers
122 views

Law of large numbers for Plancherel random Young diagrams

Do you know a reference book on the law of large numbers for random Plancherel Young diagrams ? I know the book of Kerov, but actually, it is only a compilation of his articles, and i need something ...
2
votes
2answers
148 views

Exercise at the Beginning of Part II in Fulton's Book on Young Tableaux

In Fulton's Book Young Tableaux, there's an Exercise at the beginning of part II for which I cannot find a solution (there doesn't seem to be one for this exercise in my copy of the book). It reads: ...
3
votes
1answer
171 views

Property of Young tableaux

Note: All Young diagrams are to use the English notation scheme. Suppose I have two tableaux $T$ and $T'$ on the same Young diagram (we insert the numbers $1, 2, \ldots, n$ in two different ways ...
1
vote
1answer
43 views

A question about bounding character ratios

The following question arose in a research project, and I'm sure it must be well known. I even know a very indirect proof, and would love to know if anybody knows a simple one. Here is the question. ...
5
votes
1answer
188 views

Why is every $N$-invariant polynomial function on $n\times n$ matrices in the Plücker algebra?

Let $k$ be a field and $k[{\bf x}] = k[x_{ij}: 1 \leq i, j \leq n]$ be a polynomial algebra that I can think of as the algebra of functions on $n \times n$ matrices that are polynomial in each ...
14
votes
2answers
272 views

A question on partitions of n

Let $P$ be the set of partitions of n. Let $\lambda$ denote the shape of a particular partition. Let $f_\lambda(i)$ be the frequency of $i$ in $\lambda$ and let $a_\lambda(i) := \# \lbrace j : ...
17
votes
1answer
666 views

Why is a general formula for Kostka numbers “unlikely” to exist?

In reference to Stanley's Enumerative Combinatorics Vol. 2: right after he has defined Kostka numbers (section 7.10), he mentions that it is unlikely that a general formula for $K_{\lambda\mu}$ ...