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


gives the sequence of compositions


while the semistandard generalized column tabloid


gives the sequence of compositions


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


  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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