Robinson-schensted algorithms for skew tableaux

Sagan, Bruce E.; Stanley, Richard P.

doi:10.1016/0097-3165(90)90066-6

Cited by 58 publications

(107 citation statements)

References 15 publications

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“…The technique is to apply the jeu de taquin [6,8,13,14,18] to the primed entries k of P ST taken in turn starting with any 1 s (actually there are none), then any 2 s (at most one), then any 3 s (at most two) and so on. If for fixed k there is more than one k in P ST then these are dealt with in turn from top to bottom.…”

Section: Resultsmentioning

confidence: 99%

Bijective proofs of shifted tableau and alternating sign matrix identities

Hamel

King

2006

J Algebr Comb

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We give a bijective proof of an identity relating primed shifted gl(n)-standard tableaux to the product of a gl(n) character in the form of a Schur function and 1≤i< j≤n (x i + y j ). This result generalises a number of well-known results due to Robbins and Rumsey, Chapman, Tokuyama, Okada and Macdonald. An analogous result is then obtained in the case of primed shifted sp(2n)-standard tableaux which are bijectively related to the product of a t-deformed sp(2n) character and 1≤i< j≤n (x i + t 2 x −1 i + y j + t 2 y −1 j ). All results are also interpreted in terms of alternating sign matrix (ASM) identities, including a result regarding subsets of ASMs specified by conditions on certain restricted column sums.

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Section: Resultsmentioning

confidence: 99%

Bijective proofs of shifted tableau and alternating sign matrix identities

Hamel

King

2006

J Algebr Comb

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“…Then T is obtained from P by a series of internal column insertions. Adapting the arguments of [15], we see that internal column insertion preserves Knuth equivalence. Thus P is Knuth equivalent to T. Since C is symmetric in its two arguments, Q is Knuth equivalent to U.…”

Section: 2mentioning

confidence: 88%

“…These algorithms were originally developed for tableaux of partition shape, but more recent work [4,15] uses them to define operations on pairs of skew tableaux. A striking duality between Schensted insertion and Schu tzenberger's jeu de taquin was noted by Stembridge [21] in his theory of rational tableaux.…”

Section: Introductionmentioning

confidence: 99%

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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“…Now, we recover the Knuth type correspondence for skew tableaux by Sagan and Stanley [17] as a restriction of the bijection in Theorem 5.1 to the set of LR words of shape (α, β). Theorem 5.5 Let α, β ∈ P be given.…”

Section: Remark 413mentioning

confidence: 91%

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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Robinson-schensted algorithms for skew tableaux

Cited by 58 publications

References 15 publications

Bijective proofs of shifted tableau and alternating sign matrix identities

Bijective proofs of shifted tableau and alternating sign matrix identities

Complementary Algorithms for Tableaux

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

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