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Results tagged with symmetric-groups
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user 19864
The symmetric group $S_n$ is the group of permutations of the set of integers $\{1,\dots,n\}$. This has $n!$ elements and is generated by the $n-1$ involutions exchanging consecutive integers. The symmetric groups form the simplest family of Coxeter groups.
20
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4
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An $n!\times n!$ determinant
Let us consider the matrix $A$ with its rows and columns enumerated by the elements of $S_n$ with $A_{\sigma\tau}=x^{c(\sigma\tau^{-1})}$ where $c()$ is the number of cycles in a permutation's decompo …
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24
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5
answers
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Why are Jucys-Murphy elements' eigenvalues whole numbers?
The Jucys-Murphy elements of the group algebra of a finite symmetric group (here's the definition in Wikipedia) are known to correspond to operators diagonal in the Young basis of an irreducible repre …
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13
votes
Why are Jucys-Murphy elements' eigenvalues whole numbers?
I came up with a more or less elementary proof of the identity from the top comment to my question. It involves nothing more advanced than some basic linear algebra.
Namely, let us denote $X_k=\sum\l …
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7
votes
0
answers
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Reference for an "elementary" combinatorial fact
This is a question I've been meaning to ask for quite some time.
Fact. For $n\in\mathbb N$ consider the set of segments $R=\{[i,j], 1\le i<j\le n\}$. Let a subset $E\subset R$ be nice iff $E$ is clos …
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15
votes
2
answers
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Gelfand-Tsetlin algebras and "Jucys-Murphy elements" for $\mathfrak{gl}_n$
I'm trying to figure out/find in literature the details concerning Gelfand-Tsetlin algebras for $\mathfrak{gl}_n(\mathbb C)$ (Okounkov-Vershik style, if you wish).
Consider the chain $$\mathcal U(\ma …
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4
votes
Accepted
geometric meaning to pairs of SYT indexing for the basis of cohomology ring of full flag var...
Since, surprisingly, there are still no answers or even comments, let me note that the answer to the last question is well known to be "yes": the Schubert cell containing a flag $(E_1,\dots,E_m)$ is i …
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