Questions tagged [symmetric-groups]
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.
462 questions
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Easy proof of the uncountability of bijections on natural numbers
Is there an easy proof of the uncountability of bijections on natural numbers?
The proof that I have in mind is as follows -
$\text{Gal }(\overline{\mathbb Q}/\mathbb Q)$ is a proper uncountable ...
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Function recursion relation over symmetric group
Hi!
Let P be a permutation in the symmetric group SN and let π=πj, j+1 be a transposition of elements j and j+1 of the permutation. Let A(P) be a function in dependence of the permutation P. P&...
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Hankel determinants of symmetric functions
The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions.
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Schur-Weyl duality in positive characteristic
$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SW{SW}$Let $S_k$ be the symmetric group. Let $F$ be an algebraically closed field. Let $\Rep(S_k)$ be the category of ...
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"Natural" generating sets for symmetric groups
The symmetric group on $n$ letters has
many sets of generators. Some of them are more natural than others, eg the
set $(i,i+1)$ of adjacent transpositions (natural with respect to the type A Weyl ...
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What is a concomitant (and other questions on D.E. Littlewood's "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" )?
I'm trying to understand the paper "Products and plethysms of characters with orthogonal, symplectic, and symmetric groups" by D.E. Littlewood (link), but I'm having trouble overcoming the language ...
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Where do stable Kronecker coefficients live "in nature"?
Background:
For a partition $\lambda$, let $\lambda[N] = (N - |\lambda|, \lambda_1, \lambda_2, \lambda_3, \dots)$, also let $\chi_\lambda$ be the corresponding irreducible character of the symmetric ...
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Sarrus determinant rule: references, extensions
SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS
An undergraduate came to me with an identity for 4x4 determinants that is actually correct:
$\det(A)=h(A)+h(RA)+h(R^{2}A)$
where R cyclically ...
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Algebraic structures at hypernatural parameters
Let's start with some family of algebraic structures of the same type indexed by the natural numbers, say the symmetric group $S_n$. Suppose that the axioms of this algebraic structure (in this case, ...
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Reference request: The stable Kronecker ring is isomorphic to the ring of symmetric polynomials
Background
For $\lambda$ any partition and $n$ a positive integer, write $\lambda[n]$ for the sequence $(n - |\lambda|, \lambda_1, \lambda_2, \ldots, \lambda_r)$. For $n$ large enough, this is a ...
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Explicit computation of induced modules of semidirect products with the symmetric group
I've gotten stuck in a project I have been working on, essentially on the following combinatorial question about the symmetric group.
One can obtain a 1-dimensional representation $M^n_c$ of the ...
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Is ${\rm S}_6$ the automorphism group of a group?
The automorphism group of the symmetric group $S_n$ is $S_n$ when $n$ is not $2$ or $6$, in which cases it is respectively $1$ and the semidirect product of $S_6$ with the (cyclic) group of order $2$. ...
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