## 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.

*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 two certain subsets of 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.

*Row and Column Stabilizer*

Recall that for a given -frame we defined to be the set of all Young tableaux with . What we first notice is that there is a natural -action on by permuting the numbers in the boxes. In particular,

where denotes an arbitrary element of . It’s clear that this defines an -action on as claimed.

So, suppose that we are given a tableau whose underlying Ferrer’s diagram has columns and rows are . We say that leaves a column *is setwise fixed in by * if the same numbers appear (not necessarily in the same positions) in as a column in and a column of . The meaning of a row in being *setwise fixed *is defined analogously. We then define the *column stabilizer *of some to be the set

One defines the *row stabilizer of , *denoted , similarly. It’s clear that if then since there are choices for permutations of the first row, for the second,…

Our first goal is to prove that the row and column stabilizers satisfy the relations and . But, before we point out why we consider the following lemma:

**Theorem (Conjugation Relation for Stabilizers): ***Let be a group and let be a set which has a -action denoted by concatenation. Then, for any one has that .*

**Proof: **Let then and so or and so . Conversely, if then for some and so and so . The conclusion follows.

So, how do we use this to prove our desired conjugate identities? We can do this by noting that perhaps the way we defined and maybe our action in general was the incorrect one. Namely, we could have easily defined the same actions on the inviduals rows for and the columns . But, it’s clear then from this that if the rows are of length and denote the set of subsets of of size that there is a natural action on given by for and if denotes the set of numbers in then the the number of things that fix that row under the -action on is equal to and so

Moreover, it’s clear that the set of numbers in the row of is . Thus, we see using our lemma that that

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

**Theorem: ***Let be a Young tableau with , then for each one has that*

Also from this analysis we get that

**Theorem: ***Let be any -tableau, then .*

*Stabilizer Functions*

We now define two functions associated to . Namely, we define the *row and column stabilizer functions *denoted and respectively by

where we are adding them in the sense of taking the free vector space (which of course we can identify with the group algebra by ) and is the sign function. We lastly define a third function given by (order matters, so keep that in mind).

We end this post with the following theorem:

**Theorem: ***Let be some Young tableau, then , and .*

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

**References:**

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

[…] a bijection in such a way that –it’s about time I explained roughly how. So, in our last post we created this interesting function . Our main goal to the construction is to show that up to […]

Pingback by A Weird Condition on Tableaux « Abstract Nonsense | May 22, 2011 |

[…] 1: Let be a fixed -tableaux and suppose that cannot be written as where and (the row and column stabilizers and the product is the product in not in itself). Then, there exists some and such that is […]

Pingback by Two Technical Lemmas for the Construction of the Irreps of S_n « Abstract Nonsense | May 23, 2011 |