The Unapologetic Mathematician

Mathematics for the interested outsider

Dominance for Generalized Tabloids

Sorry I forgot to post this yesterday afternoon.

You could probably have predicted this: we’re going to have orders on generalized tabloids analogous to the dominance and column dominance orders for tabloids without repetitions. Each tabloid (or column tabloid) gives a sequence of compositions, and at the ith step we throw in all the entries with value i.

For example, the generalized column tabloid

\displaystyle[S]=\begin{array}{|c|c|c|}2&1&1\\3&2&\multicolumn{1}{c}{}\end{array}

gives the sequence of compositions

\displaystyle\begin{aligned}\lambda^1&=(0,1,1)\\\lambda^2&=(1,2,1)\\\lambda^3&=(2,2,1)\end{aligned}

while the semistandard generalized column tabloid

\displaystyle[T]=\begin{array}{|c|c|c|}1&1&1\\2&3&\multicolumn{1}{c}{}\end{array}

gives the sequence of compositions

\displaystyle\begin{aligned}\mu^1&=(1,1,1)\\\mu^2&=(2,1,1)\\\mu^3&=(2,2,1)\end{aligned}

and we find that [S]\trianglelefteq[T] since \lambda^i\trianglelefteq\mu^i for all i.

We of course have a dominance lemma: if k<l, k occurs in a column to the left of l in T, and S is obtained from T by swapping these two entries, then [T]\triangleright[S]. As an immediate corollary, we find that if T is semistandard and S\in\{T\} is different from T, then [T]\triangleright[S]. That is, [T] is the "largest" (in the dominance order) equivalence class in \theta_T{t}. The proofs of these facts are almost exactly as they were before.

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February 9, 2011 - Posted by | Algebra, Representation Theory, Representations of Symmetric Groups

2 Comments »

  1. [...] would make the zero map. So among the nonzero , there are some with maximal in the column dominance order. I say that we can find a semistandard among [...]

    Pingback by Intertwinors from Semistandard Tableaux Span, part 2 « The Unapologetic Mathematician | February 12, 2011 | Reply


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