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Results tagged with young-tableaux
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user 2530
For standard Young tableaux, semistandard Young tableaux, and other related two-dimensional arrays of numbers like plane partitions. Including their combinatorial theory and their application in representation theory and algebraic geometry.
3
votes
Accepted
Flips on standard Young tableaux and descent sets
Here is a counterexample copied out of a Sage session:
sage: T = StandardTableau([[1,2,3,7,9],[4,5,8],[6,11],[10]]); T.pp()
1 2 3 7 9
4 5 8
6 11
10
sage: T.standard_descents() …
- 33.8k
6
votes
Accepted
some confusion about the explicit construction of irreducible representations of $S_n$
Yes, equivalent tableaux $t$ may yield different $e_t$'s. However, equivalent tableaux $t$ yield the equivalent $\pi t$'s for any permutation $\pi$, so that the notation $\pi\left\lbrace t\right\rbra …
- 33.8k
2
votes
What bijection on permutations corresponds under RS to transpose?
Edit: This doesn't answer the question; see comments.
Knuth, TAoCP3, p. 76, exercise 5 in section 5.1.4:
Let $P$ be the tableau corresponding to the permutation $a_1a_2...a_n$; use exercise 4 to pro …
- 33.8k
11
votes
2
answers
836
views
How exactly does Schützenberger promotion relate to Striker-Williams promotion?
Schützenberger promotion, studied (for example) in Richard Stanley, Promotion and Evacuation, 2009, is a permutation of the set of all linear extensions of a finite poset. Since one can identify the l …
- 33.8k
5
votes
1
answer
438
views
A vector version of the Segre embedding: what is the kernel of the ring map?
TL;DR version.
Given a commutative ring $\mathbf{k}$ and $n+m$ "generic" vectors $\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n, \mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_m$ in $\mathbf{k}^k$ …
- 33.8k
4
votes
Accepted
Schur functors = Weyl functors in characteristic zero?
Assume that $R$ is a field of characteristic $0$.
Recall the standard embeddings
\begin{align*}
\operatorname{Sym}^{m}V & \rightarrow V^{\otimes m},\\
v_{1}v_{2}\cdots v_{m} & \mapsto\dfrac{1}{m!}\s …
- 33.8k
14
votes
2
answers
2k
views
Sym(V ⊕ ∧² V) isomorphic to direct sum of all Schur functors of V
Let $V$ be a finite-dimensional $K$-vector space. Then, the symmetric power $\mathrm{Sym}\left(V\oplus \wedge^2 V\right)$ is isomorphic to the direct sum of all Schur functors applied to $V$ (each one …
- 33.8k
3
votes
Accepted
Proving an identity for flagged Schur without use of determinants?
Here is the argument that I was hinting to in the comments. I suspect that
this is the same argument @ZachH has in mind, but I hope to have avoided the
many opportunities to get signs and conjugates w …
- 33.8k
3
votes
On a proof involving Young symmetrizers acting on tensor spaces
I don't know what the paper is doing -- its use of Young diagrams definitely
looks vague to me, but maybe I would understand it better after reading (e.g.)
Towber. But here is an approach that avoids …
- 33.8k
6
votes
2
answers
874
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 Lec …
- 33.8k
10
votes
1
answer
680
views
Analogues of the Knuth and Forgotten equivalences on permutations: have they been studied?
Consider a totally ordered alphabet $A$ of $n$ letters. Let $W$ be the set of all words over $A$ which have no two letters equal. Then, for example, we can define the Knuth equivalence on $W$ as the s …
- 33.8k
21
votes
2
answers
2k
views
Has Reifegerste's Theorem on RSK and Knuth relations received a slick proof by now?
For the notations I am using, I refer to the Appendix at the end of this post.
Here is what, for the sake of this post, I consider to be Reifegerste's theorem:
Theorem 1. Let $n\in\mathbb N$ and $i\ …
- 33.8k
4
votes
0
answers
840
views
Categorifying the Cauchy kernel as a filtration of $\operatorname*{Sym}\left( F\otimes G\rig...
Question 1 (short version). Let $R$ be a commutative ring with unity. Let
$F$ and $G$ be two $R$-modules. Let $n\in\mathbb{N}$. Is it true that the
$n$-th symmetric power $\operatorname*{Sym}\n …
- 33.8k
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, bu …
- 33.8k
25
votes
3
answers
2k
views
Is the Ford-Fulkerson algorithm a tropical rational function?
The Ford-Fulkerson algorithm
Let me recall the standard scenario of flow optimization (for integer flows at least):
Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex s …
- 33.8k