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.
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questions with no upvoted or accepted answers
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Interpretation of "1089-number trick" in terms of symmetric group action on cohomology group?
I tried posting the following on math.stackexchange, but no answers. I can of course delete if inappropriate.
The "1089 number trick" (see e.g. here) says that if you take a three-digit ...
24
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Nekrasov-Okounkov hook length formula
I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls the Nekrasov-...
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Large values of characters of the symmetric group
For $g$ an element of a group and $\chi$ an irreducible character, there are two easy bounds for the character value $\chi(g)$: First, the bound $|\chi(g)|\leq \chi(1)$ by the dimension of the ...
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Row of the character table of symmetric group with most negative entries
The row of the character table of $S_n$ corresponding to the trivial representation has all entries positive, and by orthogonality clearly it is the only one like this.
Is it true that for $n\gg 0$, ...
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Hidden grading on $kS_n$
Brundan and Kleshchev showed that if $k$ is a field of characteristic $p$, then the group ring $kS_n$ of the symmetric group $S_n$ admits an integer grading which is nontrivial in the sense that it is ...
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Bijective proof of an identity involving number of standard Young tableaux and semistandard tableaux
Question. Can you find a bijective proof of the identity
$$ \operatorname{dim}(S^{\lambda} \mathbb{C}^m)\ \operatorname{dim}(S^{\lambda'} \mathbb{C}^n) \ f^{n^m}
= \dim \Lambda^p (\mathbb{C}^m \...
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What Is The Minimal Monomial of the Symmetric Group?
In the symmetric group $S_n$ what is the shortest sequence $c_1,\ldots,c_k\in S_n$ such that, for all $x\in S_n$ the following product of conjugates of $x$:
$$x^{c_1}x^{c_2}\ldots x^{c_k}$$
equals the ...
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A $q$-analogue of Foulkes' character related to alternating permutations
My paper "Alternating permutations and symmetric functions" at
http://math.mit.edu/~rstan/papers/altenum.pdf enumerates certain
classes of alternating permutations, such as those whose inverse is
...
11
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249
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Which irreducible representations of the symmetric group are eigenspaces of class sums?
In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
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Has anyone met this "$q$-character" table for $S_4$?
Is anyone aware of the following $q$-character table for the
symmetric group $S_4$?
\begin{array}{|c|c|c|c|c|c|}
\hline
\mathrm{conj}\backslash\mathrm{rep}
& 2+1+1 & 3+1 & ...
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Smith normal form of conjugacy class actions
This question was inspired by Smith Normal Form of a Cayley Graph of the Symmetric Group.
Let $\mathbb{Q}S_n$ denote the group algebra over $\mathbb{Q}$ of the
symmetric group $S_n$. Identify a ...
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An identity for characters of the symmetric group
I am looking for a reference for the identity
$$\chi_\lambda(C)=\frac{\dim(V_\lambda)}{|C|}\sum_{p\in P_\lambda,\,q\in Q_\lambda,\,pq\in C}\operatorname{sgn}(q)$$
for the irreducible characters of the ...
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Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule
$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
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A question related to Young symmetrizers
Let $T$ be an arbitrary Young tableau (i.e., filling of the diagram of an integer partition $\lambda$ of $n$ by the numbers from $1$ to $n$, each appearing once). Let $R(T)$ be the subgroup of ...
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Generalized Young symmetrizers, why not?
For $n$ a positive integer, let $[n]=\{1,2,\ldots,n\}$. Consider two set partitions $\mathcal{A}=\{A_1,\ldots,A_p\}$ and $\mathcal{B}=\{B_1,\ldots,B_q\}$ of the set $[n]$.
We will denote by $G(\...
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Non-zero group determinant for symmetric group
Let $G$ be a finite group. Given complex numbers $x=\{x_g: g\in G\}$, one can define a $|G|\times |G|$ matrix $X$, with entries $X_{g,h} = x_{gh^{-1}}$.
Let's consider $G$ being the symmetric group $...
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The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial
I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...
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Optimizing computations with nilpotents in a group algebra
Of course, I have a very concrete problem at hand, which has been vexing me for about a year now. But let me start with a question that has a better chance of having been answered.
Let $G$ be a ...
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Explaining $\left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(a^i-1\right)$ by a free $S_n$-action
Here is an olympiad-level problem on elementary number theory:
Let $a$ be an integer and $n$ a positive integer. Prove that
\begin{align}
\left(a-1\right)^n \cdot n! \mid a^{n-1} \prod_{i=1}^n \left(...
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Classification of posets that are quotient posets of the Boolean lattice
Quotient posets of the Boolean lattice $B_n$ have interesting properties and are for example discussed in chapter 5 of Stanley's book on algebraic combinatorics.
$B_n/G$ for a subgroup $G$ of the ...
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A question regarding an analog of Young symmetrizer: the product row and column preserving subgroups without sign representation
Consider a rectangular Young diagram $\lambda$ with $n = pq$ boxes, with $p$ rows and $q$ columns. If $C$ is the column preserving subgroup of $\lambda$ and $R$ is the row preserving subgroup, then we ...
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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 ...
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498
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Representation theory of symmetric group for dummies
I have to grade a master project on representations of symmetric groups (char $0$) third time in my life and finally I came to a conclusion that I have to get a grasp of the matter. I am aware of ...
6
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The meet of two dominant permutations in weak order of $S_n$
A permutation is called dominant if its Lehmer code is a partition, or equivalently if it avoids the pattern $132$.
I can prove that given a permutation $v\in S_n$, there is a unique dominant ...
6
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What about Hopf algebra and fusion structures for intertwiner algebras?
Let $G$ be a complex, reductive group and let $V_1, \dotsc, V_r$ be a collection of finite dimensional, irreducible complex
representations of $G$. Let $\mathcal{A} = \mathrm{End}_G(V_1 \otimes \dotsb ...
6
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Proving the spectrum of the Young-Jucys-Murphy elements by formal computation in the degenerate affine Hecke algebra
This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...
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What do Macdonald polynomials hint about $\operatorname{Rep}(S_\infty)$?
$\DeclareMathOperator\Rep{Rep}$It is well-known that the Macdonald "$P$" polynomials deform the
Jack "$J$" polynomials [1]. The latter have profound relations with
representation ...
6
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Combinatorics of $p$-Kazhdan--lusztig polynomials
When can we (and can we not!) understand the dimensions of simple modules, $D(\lambda)$, of symmetric groups in a combinatorial fashion?
Let's assume that I'm going to try to do this using the theory ...
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Natural maps between Schur functors: understanding the image
Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map
$$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$
Let $[\Lambda^2 V]...
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Permanent of the symmetric group
Let $A$ be the algebra corresponding to a representation-finite block of a Schur algebra.
See for example 6.1. of https://arxiv.org/pdf/1607.05965.pdf for quiver and relations and some relevance of ...
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hooks and contents: Part II
This is a 2nd installment to my earlier MO question for which Mark Wildon furnished a clean answer.
$\mathcal{O}(\pi)$ and $\mathcal{E}(\pi)$ stand for the number of odd and even cycles of a ...
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A particular proof of the Littlewood Richardson rule
Given $\lambda \subseteq \nu$ we define a tableau of shape $\nu\setminus \lambda$ and weight $\mu$ to be a map ${\sf T}: [\nu\setminus\lambda] \rightarrow \{1,\ldots, r\}$ such that $\mu_c=|\{ x \...
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Finding a maximal tableau for a sum of Jucys-Murphy Elements
Let $X \in \mathbb{C}[S_n]$ be an element of the group algebra of $S_n$ expressible as the sum of some Jucys-Murphy elements. Then let $\lambda$ be any irreducible representation of $S_n$, with the ...
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Subgroups of the symmetric group and binary relations
Motivation
The following came up in my work recently. (NB this is the motivation, not the question I'm asking. You can skip to the actual question below, which is self-contained, but not self-...
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Towers of algebras, their 2-step centralizer algebras, and analogues of the degenerate affine Hecke algebra
Let $\, \big( {\frak{F}}_0 \subset {\frak{F}}_1 \subset {\frak{F}}_2 \subset \cdots \big)$ be a tower of semi-simple, finite dimensional, unital, complex algebras starting with ${\frak{F}}_0 \cong \...
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Finite trees with forests realizing all partitions
Removing interiors of some edges in a tree with $n$ vertices leaves a spanning-forest
with $k$ connected components (given by subtrees) having respectively $\lambda_1,\ldots,\lambda_k$
vertices. We ...
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answers
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Narayana numbers as character values?
The Catalan numbers show up as character values of the symmetric group: Let $\lambda = (n,n)$, a partition with two parts. Then $\chi^\lambda(1^{2n}) = \frac{1}{n+1}\binom{2n}{n}$, the $n$:th Catalan ...
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Tabloid Construction of Permutation Representation of Hyperoctahedral Group
For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\...
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What are the relation between Rep(G) and Rep(S_n)?
Let G be a finite group. We know it can be written as a subgroup of S_n. On the other hand, people sometimes say Rep(G) --- the category of all finite dimensional representations, are more interesting ...
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Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram
I have a "very" indirect proof that the following fact is true for every Young diagram $\lambda \vdash n$ and every $r \in \{0,\dotsc,n\}$:
\begin{equation}
d_\lambda = \sum_{a \in \mathrm{...
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What is $\dim D^{\lambda}$ for the symmetric group?
What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...
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Detecting symmetries in polynomials that lead to nice geometric properties
If we plot the single variable polynomial $p(x) = (x^2-1)^2$, it is easy to see that it has a nice property: all of its local minima are actually global minima.
In particular, it has precisely two ...
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Words that give rise to an enumeration of elements of the symmetric group
Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration ...
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Question about generalizing Cauchy identity
One of the Cauchy identities says that
$$\prod_{i,j}(1+x_iy_j) = \sum_\lambda s_\lambda (x_1, \cdots,x_m) s_{\lambda'}
(y_1, \cdots,y_n) $$
Where $\lambda$ is a Young diagram, $\lambda'$ is the ...
4
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Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
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Ref. request: Enumerating elements of Bruhat cells
Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then
$$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$
where we embed the ...
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Characterizing the RSK corespondance
The Robinson-Schensted-Knuth correspondence is an algorithm which takes as input a word $w$ on the alphabet $\{1,\dots,n\}$ of length $k$ and returns a pair of a tableau $P(w)$ and a standard tableau $...
4
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Permutation-invariant matrix representation
The question guide says that Mathoverflow is for research level mathematics. While I do not perform research in mathematics (I study quantum chemistry), I believe this question is research-level ...
4
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Name for class of flattening permutations
Let $S_n$ be the symmetric group. For any sequence of numbers $y=[y_1,y_2,\cdots,y_k]$, define the flattening operation as $\mbox{flatt}_{k}(y)$ as a relabeling of $y_1,y_2,\cdots,y_k$ in terms of ...
4
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Partial order on partitions and symmetric group algebra
Let $n$ be a natural number. Consider a set $\Lambda_n$ of partitions of $n$ into a sum of natural numbers, like $n = \lambda_1 + \cdots +\lambda_k$ (A set of small lambdas representing a partition is ...