The Symmetric Group

Abstract. The descent set of an oscillating (or up-down) tableau is introduced. This descent set plays the same role in the representation theory of the symplectic groups as the descent set of a standard tableau plays in the representation theory of the general linear groups. In particular, we show that the descent set is preserved by Sundaram's correspondence. This gives a direct combinatorial interpretation of the branching rules for tensor products of the defining representation of the symplectic groups; equivalently, for the Frobenius character of the action of a symmetric group on an isotypic subspace in a tensor power of the defining representation of a symplectic group.

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“…For a textbook treatment we refer to the books of Sagan [10,Chapter 3] or Stanley [12,Chapter 7]. Let V be the defining representation of GL(n).…”

Section: Robinson-schensted and The General Linear Groupsmentioning

confidence: 99%

“…. w r in the crystal graph corresponding to ⊗ r V has a descent at position k if there is a (nontrivial) directed path from w k to w k+1 in the crystal graph (10).…”

Section: Oscillating Tableaux and Descentsmentioning

confidence: 99%

Descent sets for symplectic groups

2013

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“…A classical result of the algorithm is the symmetry property: Theorem 1 (See e.g. [9,21,23]) For any permutation σ ,…”

Section: Symmetry Property For the Robinson-schensted Algorithmmentioning

confidence: 99%

A symmetry property for $$q$$ q -weighted Robinson–Schensted and other branching insertion algorithms

Pei

2014

J Algebr Comb

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In [19], a q-weighted version of the Robinson-Schensted algorithm was introduced. In this paper, we show that this algorithm has a symmetry property analogous to the well-known symmetry property of the usual Robinson-Schensted algorithm. The proof uses a generalisation of the growth diagram approach introduced by Fomin [5][6][7][8]. This approach, which uses 'growth graphs', can also be applied to a wider class of insertion algorithms which have a branching structure, including some of the other q-weighted versions of the Robinson-Schensted algorithm which have recently been introduced by .

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“…We define these and related partial orders as follows. (For more information see, e.g., [21], [52], [54], [55].) Call a weakly decreasing sequence λ = (λ 1 , .…”

Section: A Multigrading Of C[x] and Two Basesmentioning

confidence: 99%

“…As an application, consider the irreducible character immanants defined [34, where χ λ : S n → C is the irreducible S n -character corresponding to λ. (See, e.g., [52].) Merris [44,Cor.…”

Section: One Partial Filtration Ofmentioning

confidence: 99%

Bitableaux and Zero Sets of Dual Canonical Basis Elements

Rhoades

Skandera

2011

Ann. Comb.

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Abstract. We state new results concerning the zero sets of polynomials belonging to the dual canonical basis of C[x 1,1 , . . . , x n,n ]. As an application, we show that this basis is related by a unitriangular transition matrix to the simpler bitableau basis popularized by Désarménien-Kung-Rota. It follows that spaces spanned by certain subsets of the dual canonical basis can be characterized in terms of products of matrix minors, or in terms of their common zero sets.

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The Symmetric Group

Cited by 692 publications

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Descent sets for symplectic groups

Descent sets for symplectic groups

A symmetry property for $$q$$ q -weighted Robinson–Schensted and other branching insertion algorithms

Bitableaux and Zero Sets of Dual Canonical Basis Elements

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