Abstract Nonsense

Crushing one theorem at a time

Row and Column Stabilizer


Point of Post: In this post we define the notion of the column and row stabilizers for a Young tableaux and some standard results. Of course we’ll have to talk about the appropriate action first.

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Motivation

We now start to move away from pure combinatorics we’ve been engaging in and start to prepare for the representation theory that lies ahead. But, before we get into the pure rep theory we need to start with a mix of algebra and combinatorics to start. Roughly, in this post we define for each tableau \mathcal{T} two certain subsets of S_n that ‘stabilizes’ it in a particular interesting way. We then consider certain sums and products in the group algebra associated to these two certain subsets.

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Row and Column Stabilizer

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Recall that for a given n-frame \mathcal{F} we defined \text{Tab}\left(\mathcal{F}\right) to be the set of all Young tableaux \mathcal{T} with \text{Frame}\left(\mathcal{T}\right)=\mathcal{F}. What we first notice is that there is a natural S_n-action on \text{Tab}\left(\mathcal{F}\right) by permuting the numbers in the boxes. In particular,

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\pi\begin{array}{cc} \quad\,\,\,\begin{array}{|c|c|}\cline{1-2} * & *\\ \cline{1-2}\end{array} & \cdots\\ \begin{array}{c}\begin{array}{|c|}\cline{1-1} *\\ \cline{1-1}\end{array}\\ \vdots\end{array} & \ddots\end{array}=\begin{array}{cc} \quad\;\;\;\;\;\,\,\begin{array}{|c|c|}\cline{1-2} \pi(*) & \pi(*)\\ \cline{1-2}\end{array} & \cdots\\ \begin{array}{c}\begin{array}{|c|}\cline{1-1} \pi(*)\\ \cline{1-1}\end{array}\\ \vdots\end{array} & \ddots\end{array}

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where \ast denotes an arbitrary element of [n]. It’s clear that this defines an S_n-action on \text{Tab}\left(\mathcal{F}\right) as claimed.

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So, suppose that we are given a tableau \mathcal{T} whose underlying Ferrer’s diagram has columns C_1,\cdots,C_p and rows are R_1,\cdots,R_q. We say that leaves a column C_k is setwise fixed in \mathcal{T} by \pi\in S_n if the same numbers appear (not necessarily in the same positions) in C_k as a column in \mathcal{T} and a column of \pi\mathcal{T}. The meaning of a row R_k in \mathcal{T} being setwise fixed is defined analogously. We then define the column stabilizer of some \mathcal{T} to be the set

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\mathcal{C}\left(\mathcal{T}\right)=\left\{\pi\in S_n:\text{ every column is left setwise fixed in }\mathcal{T}\text{ by }\pi\right\}

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One defines the row stabilizer of \mathcal{T}, denoted \mathcal{R}\left(\mathcal{T}\right), similarly. It’s clear that if \text{Frame}\left(\mathcal{T}\right)=(m_1,\cdots,m_r) then \#\left(\mathcal{R}\left(\mathcal{T}\right)\right)=m_1!\cdots m_r! since there are m_1! choices for permutations of the first row, m_2! for the second,…

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Our first goal is to prove that the row and column stabilizers satisfy the relations \mathcal{C}(\pi \mathcal{T})=\pi\mathcal{C}\left(\mathcal{T}\right)\pi^{-1} and \mathcal{R}\left(\pi\mathcal{T}\right)=\pi\mathcal{R}\left(\mathcal{T}\right)\pi^{-1}. But, before we point out why we consider the following lemma:

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Theorem (Conjugation Relation for Stabilizers): Let G be a group and let X be a set which has a G-action denoted by concatenation. Then, for any x\in X one has that \text{stab}(gx)=g\text{ stab}(x)g^{-1}.

Proof: Let h\in\text{stab}(gx) then (hg)(x)=h(gx)=gx and so (g^{-1}hg)x=x or g^{-1}hg\in\text{stab}(x) and so h\in g\text{ stab}(x)g^{-1}. Conversely, if h\in g\text{ stab}(x)g^{-1} then h=gsg^{-1} for some s\in\text{stab}(x) and so h(gx)=(gsg^{-1}g)x=gsx=gx and so h\in\text{stab}(gx). The conclusion follows. \blacksquare

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So, how do we use this to prove our desired conjugate identities? We can do this by noting that perhaps the way we defined \mathcal{R}\left(\mathcal{T}\right),\mathcal{C}\left(\mathcal{T}\right) and maybe our action in general was the incorrect one. Namely, we could have easily defined the same actions on the inviduals rows R_1,\cdots,R_q for \mathcal{T} and the columns C_1,\cdots,C_p. But, it’s clear then from this that if the rows are of length m_1,\cdots,m_q and \displaystyle {[n]\choose m_k} denote the set of subsets of [n] of size m_k that there is a natural action on \displaystyle {[n] \choose m_k} given by \pi A=\pi(A) for \displaystyle A\in{[n]\choose m_k} and if S_k denotes the set of numbers in R_k then the the number of things that fix that row under the S_n-action on \text{Tab}\left(\mathcal{F}\right) is equal to \text{stab}\left(S_k\right) and so

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\displaystyle \mathcal{R}\left(\mathcal{T}\right)=\bigcap_{k=1}^{q}\text{stab}\left(S_k\right)

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Moreover, it’s clear that the set of numbers in thek^{\text{th}} row of \pi\mathcal{T} is \pi S_k. Thus, we see using our lemma that that

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\displaystyle \mathcal{R}\left(\pi\mathcal{T}\right)=\bigcap_{k=1}^{q}\text{stab}\left(\pi S_k\right)=\bigcap_{k=1}^{q}\pi\text{ stab}(S_k)\pi^{-1}=\pi\bigcap_{k=1}^{q}\text{stab}(S_k)\pi^{-1}=\pi\mathcal{R}\left(\mathcal{T}\right)\pi^{-1}

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and since a similar method works for the column stabilizer we arrive at:

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Theorem: Let \mathcal{T} be a Young tableau with \text{Frame}\left(\mathcal{T}\right)\in\text{Frame}_n, then for each \pi\in S_n one has that

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\mathcal{R}\left(\pi\mathcal{T}\right)=\pi\mathcal{R}\left(\mathcal{T}\right)\pi^{-1}\quad\text{and}\quad\mathcal{C}\left(\pi\mathcal{T}\right)=\pi\mathcal{C}\left(\mathcal{T}\right)\pi^{-1}

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Also from this analysis we get that

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Theorem: Let \mathcal{T} be any n-tableau, then \mathcal{R}\left(\mathcal{T}\right),\mathcal{C}\left(\mathcal{T}\right)\leqslant S_n.

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Stabilizer Functions

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We now define two functions associated to \mathcal{R}\left(\mathcal{T}\right),\mathcal{C}\left(\mathcal{T}\right). Namely, we define the row and column stabilizer functions denoted P\left(\mathcal{T}\right) and Q\left(\mathcal{T}\right) respectively by

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\displaystyle P\left(\mathcal{T}\right)=\sum_{\pi\in\mathcal{R}\left(\mathcal{T}\right)}\pi\quad\text{and}\quad Q\left(\mathcal{T}\right)=\sum_{\pi\in\mathcal{C}\left(\mathcal{T}\right)}\text{sgn}(\pi)\pi

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where we are adding them in the sense of taking the free vector space \mathbb{C}\left[S_n\right] (which of course we can identify with the group algebra by \pi\leftrightarrow \delta_\pi) and \text{sgn} is the sign function.  We lastly define a third function E:\text{Tab}\left(\mathcal{F}\right)\to\mathbb{C}\left[S_n\right] given by E\left(\mathcal{T}\right)=P\left(\mathcal{T}\right)Q\left(\mathcal{T}\right) (order matters, so keep that in mind).

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We end this post with the following theorem:

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Theorem: Let \mathcal{T} be some Young tableau, then P\left(\pi\mathcal{T}\right)=\pi P\left(\mathcal{T}\right)\pi^{-1}, Q\left(\pi\mathcal{T}\right)=\pi Q\left(\mathcal{T}\right)\pi^{-1} and E\left(\pi\mathcal{T}\right)=\pi E\left(\mathcal{T}\right)\pi^{-1}.

Proof: The first two of these follow from previous theorems and the last follows immediately from the first two. \blacksquare

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References:

1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

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May 20, 2011 - Posted by | Algebra, Algebraic Combinatorics, Representation Theory | , , , , ,

2 Comments »

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