Insertion and the multiplication of rational schur functions

Stroomer, Jeffrey

doi:10.1016/0097-3165(94)90039-6

Cited by 8 publications

(9 citation statements)

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“…For a combinatorial realization, most crystals in this paper are embedded in a set of binary matrices of various shapes, equivalently an (infinite) abacus model. Also, two kinds of Kashiwara operators on binary matrices [4,17,20], which produces various dualities, play a crucial role in proving our main results, while the rational semistandard tableaux for gl n [25,26] were used to understand extremal weight crystals of type A +∞ [18].…”

Section: This Work Was Supported By Lg Yonam Foundationmentioning

confidence: 99%

Crystal duality and Littlewood–Richardson rule of extremal weight crystals

Kwon

2011

Journal of Algebra

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We consider a category of gl ∞ -crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight gl ∞ -modules with respect to derivations shifting the characters of fundamental weight modules. A Littlewood-Richardson rule of extremal weight crystals of non-negative level is described explicitly in terms of classical Littlewood-Richardson coefficients.

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Section: This Work Was Supported By Lg Yonam Foundationmentioning

confidence: 99%

Crystal duality and Littlewood–Richardson rule of extremal weight crystals

Kwon

2011

Journal of Algebra

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“…Either an entry of P C will be bumped from P C or this reverse insertion will result in a new box on the inner border of P C . Stroomer [22] showed that in the first case, it is not permissible to begin a column slide at b on P, and in the second case, the column slide is permissible, and T C is the tableau obtained from the (internal) column insertion. Thus P C and T C are Knuth equivalent, and so by Theorem 2.2, P and T are Knuth equivalent.…”

Section: Jeux De Tableauxmentioning

confidence: 98%

“…In combinatorial representation theory, complementation provides a combinatorial model for the procedure of dividing by the determinantal representation of GL k . In this context, it was used by Stembridge [21] and Stroomer [22] to develop combinatorial algorithms for studying rational representations of GL k . (The classical Robinson-Schensted-Knuth correspondence [11,16,6] is used to study polynomial representations of GL k .)…”

Section: Complementationmentioning

confidence: 99%

“…Column insertion [3,14] is another tableau algorithm that can be extended to define the operation of internal column insertion between pairs of tableaux. The operation hopscotch which is complementary to internal column insertion is new, although it extends both Stroomer's algorithm of column-sliding [22] and Tesler's [23] rightward-and leftward-shift games.…”

Section: Introductionmentioning

confidence: 99%

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Complementary Algorithms for Tableaux

Roby

Sottile

Stroomer

et al. 2001

Journal of Combinatorial Theory, Series A

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We study four operations defined on pairs of tableaux. Algorithms for the first three involve the familiar procedures of jeu de taquin, row insertion, and column insertion. The fourth operation, hopscotch, is new, although specialised versions have appeared previously. Like the other three operations, this new operation may be computed with a set of local rules in a growth diagram, and it preserves the Knuth equivalence class. Each of these four operations gives rise to an a priori distinct theory of dual equivalence. We show that these four theories coincide. The four operations are linked via the involutive tableau operations of complementation and conjugation. Academic Press

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“…Let us review an insertion algorithm for B µ,ν [10], which is an infinite analogue of those for rational semistandard tableaux for gl n [17,18]. For a ∈ N and (S, T ) ∈ B µ,ν , we define a → (S, T ) in the following way;…”

Section: And J < L)mentioning

confidence: 99%

A plactic algebra of extremal weight crystals and the Cauchy identity for Schur operators

Kwon

2011

J Algebr Comb

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We give a new bijective interpretation of the Cauchy identity for Schur operators which is a commutation relation between two formal power series with operator coefficients. We introduce a plactic algebra associated with the Kashiwara's extremal weight crystals over the Kac-Moody algebra of type A +∞ , and construct a Knuth type correspondence preserving the plactic relations. This bijection yields the Cauchy identity for Schur operators as a homomorphic image of its associated identity for plactic characters of extremal weight crystals, and also recovers Sagan and Stanley's correspondence for skew tableaux as its restriction.Keywords Plactic algebra · Crystal · Schur operator where s λ and s ⊥ λ are linear operators on Λ induced from the left multiplication by s λ (x) and its adjoint with respect to the Hall inner product on Λ, respectively. One may regard s λ and s ⊥ λ as operators on QP = λ∈P Qλ, where λ is identified with 428 J Algebr Comb (2011) 34:427-449 s λ (x). Moreover s λ and s ⊥ λ can be given as Schur functions in certain locally noncommutative operators on QP called Schur operators by Fomin, while P (x) and Q(x) can be written as Cauchy products in Schur operators and x [3,4].Let y = {y 1 , y 2 , . . .} be another set of formal commuting variables. It is well known that the following commutation relation holds:1.1) called generalized Cauchy identity or Cauchy identity for Schur operators. Considering both sides as operators with coefficients in Λ x ⊗ Λ y and then equating each entry of their matrix forms, we obtain a Cauchy identity for skew Schur functions [16],where α, β are given partitions. A bijective interpretation of the Cauchy identity for skew Schur functions was given by Sagan and Stanley [17], and it was extended to a bijection in a more general framework by Fomin [3] including various analogues of Knuth correspondence.Recently, a new representation theoretic interpretation of the Cauchy identity for Schur operators was given by the author [11] using the notion of Kashiwara's extremal weight crystals [8] over the quantized enveloping algebra associated with the Kac-Moody algebra of type A +∞ , say gl >0 . It is proved that a Schur operator can be realized as a functor of tensoring by an extremal weight crystal element and (1.1) can be understood as a non-commutative character identity corresponding to the decomposition of the crystal graph of the Fock space with infinite positive level, which is an infinite analogue of the level n fermionic Fock space decomposition due to Frenkel [5].Motivated by a categorification of Schur operators in [11], we give a new combinatorial way to explain both the Cauchy identities for Schur operators and skew Schur functions in terms of a single bijection. More precisely, the main result in this paper is to construct a Knuth type correspondence, which gives a bijective interpretation of the identity (1.1) or its dual form, as the usual Knuth correspondence does for the Cauchy product, and also recovers the Sagan and Stanley's correspondence as its restrictio...

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Insertion and the multiplication of rational schur functions

Cited by 8 publications

References 4 publications

Crystal duality and Littlewood–Richardson rule of extremal weight crystals

Crystal duality and Littlewood–Richardson rule of extremal weight crystals

Complementary Algorithms for Tableaux

A plactic algebra of extremal weight crystals and the Cauchy identity for Schur operators

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