Orthogonal tableaux and an insertion algorithm for SO(2n + 1)

Sundaram, Sheila

doi:10.1016/0097-3165(90)90059-6

Cited by 61 publications

(63 citation statements)

References 6 publications

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“…This was notably realized by Berele in [1] for sp 2n , by Okada in [21] for so 2n and by Sundaram in [24] for so 2n+1 (see also [6] for a complete list of references). Nevertheless, it seems difficult to derive plactic relations from these schemes and to relate them to the Kashiwara crystal basis theory.…”

Section: Introductionmentioning

confidence: 99%

Crystal bases and combinatorics of infinite rank quantum groups

Lecouvey

2008

Trans. Amer. Math. Soc.

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Abstract. The tensor powers of the vector representation associated to an infinite rank quantum group decompose into irreducible components with multiplicities independent of the infinite root system considered. Although the irreducible modules obtained in this way are not of highest weight, they admit a crystal basis and a canonical basis. This permits us in particular to obtain for each family of classical Lie algebras a Robinson-Schensted correspondence on biwords defined on infinite alphabets. We then depict a structure of bicrystal on these biwords. This RSK-correspondence yields also a plactic algebra and plactic Schur functions distinct for each infinite root system. Surprisingly, the algebras spanned by these plactic Schur functions are all isomorphic to the algebra of symmetric functions.

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Section: Introductionmentioning

confidence: 99%

Crystal bases and combinatorics of infinite rank quantum groups

Lecouvey

2008

Trans. Amer. Math. Soc.

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“…The charaters so 2n+1 (λ, X ) can be interpreted in terms of a set of orthogonal tableau of shape λ, as denoted by O (λ) and introduced by Sundaram [39]. The Proctor tableaux [35] also leads to the same character as the Sundaram tableaux, and a weight preserving bijection of these two classes of tableaux is established by Fulmek and Krattenthaler [8].…”

Section: The Odd Orthogonal Charactersmentioning

confidence: 99%

“…We find another application of this idea to the symplectic and orthogonal characters sp(2n) and so(2n + 1) by giving new flagged determinantal formulas for these two kinds of characters. They have been studied via various approaches, see, for example, [11,39]. Fulmek and Krattenthaler [11] give a proof for the determinant expression…”

Section: Flagged Determinantal Formulas For Sympletic and Orthogonal mentioning

confidence: 99%

“…Finally, we obtain lattice path representations of the tableau definitions of the symplectic and orthogonal characters of sp 2n (λ, X ) and so 2n+1 (λ, X ) based on the tableau representations of King and El-Sharkaway [20], and Sundaram [39]. Based on such lattice path correspondence, we obtain two flagged determinantal formulas for these characters.…”

Section: Introductionmentioning

confidence: 99%

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Chen

Louck

2002

Journal of Algebraic Combinatorics

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Abstract. The double Schur function is a natural generalization of the factorial Schur function introduced byBiedenharn and Louck. It also arises as the symmetric double Schubert polynomial corresponding to a class of permutations called Grassmannian permutations introduced by A. Lascoux. We present a lattice path interpretation of the double Schur function based on a flagged determinantal definition, which readily leads to a tableau interpretation similar to the original tableau definition of the factorial Schur function. The main result of this paper is a combinatorial treatment of the flagged double Schur function in terms of the lattice path interpretations of divided difference operators. Finally, we find lattice path representations of formulas for the symplectic and orthogonal characters for sp(2n) and so(2n + 1) based on the tableau representations due to King and El-Shakaway, and Sundaram. Based on the lattice path interpretations, we obtain flagged determinantal formulas for these characters.

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“…Although the Pieri rule for GL n is equivalent to the GL n to GL n−1 branching rule, the analogs for Sp 2n and SO n are noticeably more complicated than the corresponding branching rules. There is a description of such tensor products in terms of tableaux ([Su1,Su2]), but it involves erasing boxes as well as adding boxes, making descriptions of iterated tensor products somewhat awkward. However, two of the present authors ( [KL]) have found that an algebra naturally associated to the Pieri-like tensor products for Sp 2n and SO n , is a deformation of a Hibi ring, at least under a certain stability condition.…”

Section: 4mentioning

confidence: 99%

Double Pieri algebras and iterated Pieri algebras for the classical groups

Howe¹,

Kim²,

Lee³

2017

American Journal of Mathematics

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Abstract. We study iterated Pieri rules for representations of classical groups. That is, we consider tensor products of a general representation with multiple factors of representations corresponding to one-rowed Young diagrams (or in the case of the general linear group, also the duals of these). We define iterated Pieri algebras, whose structure encodes the irreducible decompositions of such tensor products. We show that there is a single family of algebras, which we call double Pieri algebras, and which can be used to describe the iterated Pieri algebras for all three families of classical groups. Furthermore, we show that the double Pieri algebras have flat deformations to Hibi rings on explicitly described posets.

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Orthogonal tableaux and an insertion algorithm for SO(2n + 1)

Cited by 61 publications

References 6 publications

Crystal bases and combinatorics of infinite rank quantum groups

Crystal bases and combinatorics of infinite rank quantum groups

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Double Pieri algebras and iterated Pieri algebras for the classical groups

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