All Questions
Tagged with algebraic-combinatorics symmetric-groups
12
questions
2
votes
0
answers
81
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Restriction of induced representation over a Young subgroup and Littlewood-Richardson coefficients
I'm inexperienced in the representation theory of the symmetric group, so please correct my possible mistakes. Fix $m\leq n$, $G:=S_n$ and $H:=S_m\times S_{n-m}$ as a Young subgroup of $G$. Let $V^{\...
- 4,886
0
votes
0
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107
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Hall Scalar product calculation: $\langle h_3p_6, s_{(5,4)}\rangle$
This is a problem from an old qualifying exam I am reviewing:
Calculate: $\langle h_3p_6, s_{(5,4)}\rangle$ where $\langle \cdot,\cdot \rangle$ is the Hall scalar product defined by $\langle s_\lambda,...
0
votes
0
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190
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Decomposing into irreducible $S_n$ modules, aka Specht modules.
Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$.
I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...
- 2,591
3
votes
0
answers
105
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Combinatorial bijection of primitive factorization
Let $\nu$ is a partition on $n$. Given a $n$ cycle $(1,2,\ldots,n)\in S_n$.
Let define $H_g^{m}((n);\mu)$ count the number of tuples $(\tau_1,\ldots,\tau_r)$ in symmetric group $S_n$.Let $\beta$ ...
- 1,065
1
vote
0
answers
64
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Combinatorial proof of the identity relating Hurwitz numbers
Let define $H^{m}((n);\mu)$ count the number of tuples $(\alpha,\tau_1,\ldots,\tau_r ,\beta)$ in symmetric group $S_n$ where $\alpha$ is fixed cycle of type $(n)$ and $\beta$ cycle of type $\mu$ and
...
- 1,065
2
votes
0
answers
107
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Hook-length formula [closed]
Let $\lambda=(\lambda_1,\ldots,\lambda_n)$ be a partition of $d$. Then hook length formula gives us
$$dim(\lambda)=\chi_{1^d}^{\lambda}=\frac{d!}{\prod h_{\lambda}(i,j)}$$
where $\chi_{a}^{b}$ denote ...
- 1,065
9
votes
1
answer
746
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Young projectors in Fulton and Harris
In Section 4 of Fulton and Harris' book Representation Theory, they give the definition of a Young tableau of shape $\lambda = (\lambda_1,\dots,\lambda_k)$ and then define two subgroups of $S_d$, the ...
- 25.2k
1
vote
0
answers
96
views
Murnaghan-Nakayama rule for general dimension of a hook
Let $m,r\in \mathbb{N}, n=rm$. The character of symmetric group $\chi^{\lambda}$ where $\lambda$ is of the from
$(k,1^{n-k})$ evaluated at the conjugacy class of $S_{n}$ of the from $(r,\ldots,r)$, ...
- 1,065
1
vote
0
answers
134
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Permutations That Are Conjugate with an Element From Stabilizer of Another Permutation
We know that permutations, elements of the symmetric group on a finite set with n elements, are conjugate iff they have the same cycle structure.
My question is that given two permutations that are ...
- 149
2
votes
1
answer
1k
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nth power symmetric polynomial in terms of Schurs polynomial
The Schur's polynomial forms the basis of the symmetric algebra so does the power symmetric function. nth power symmetric function are the function of the form $\sum_i x_i^n$. Let $\lambda \vdash n$ ...
- 1,065
5
votes
2
answers
291
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Is there a synthetic definition of the $0$-Hecke monoid of $S_n$?
Background. Let $n$ be a nonnegative integer, and let $S_n$ denote the $n$-th symmetric group. The $0$-Hecke monoid $H_0\left(S_n\right)$ is defined to be the monoid given by generators $t_1, t_2, \...
- 17.9k
3
votes
0
answers
234
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Symmetrization of Powersum polynomials
Let $n\in\mathbb{N}$. Then, for $i\in\mathbb{N}$, the $i-$th power sum if defined to be the polynomial $p_i^{(n)}:=\sum_{j=1}^n x_j^i$ in $n$ indeterminates $x_1,x_2,\ldots,x_n$.
Then let $\lambda:=(\...
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