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The shifted plactic monoid
, 2008
"... We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) th ..."
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We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted LittlewoodRichardson Rule; similar results for the coefficients in the Schur expansion of a Schur Pfunction; a shifted counterpart of the LascouxSchützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.
A variation on tableau switching and a PakVallejo’s Conjecture, DMTCS proc
 FPSAC /SFCA 2008
, 2008
"... Abstract. Pak and Vallejo have defined fundamental symmetry map as any Young tableau bijection for the commutativity of the LittlewoodRichardson coefficients cλµ,ν = c λ ν,µ. They have considered four fundamental symmetry maps and conjectured that they are all equivalent (2004). The three first on ..."
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Abstract. Pak and Vallejo have defined fundamental symmetry map as any Young tableau bijection for the commutativity of the LittlewoodRichardson coefficients cλµ,ν = c λ ν,µ. They have considered four fundamental symmetry maps and conjectured that they are all equivalent (2004). The three first ones are based on standard operations in Young tableau theory and, in this case, the conjecture was proved by Danilov and Koshevoy (2005). The fourth fundamental symmetry, given by the author in (1999;2000) and reformulated by Pak and Vallejo, is defined by nonstandard operations in Young tableau theory and will be shown to be equivalent to the first one defined by the involution property of the BenkartSottileStroomer tableau switching. The proof of this equivalence provides, in the case the first tableau is Yamanouchi, a variation of the tableau switching algorithm which shows switching as an operation that takes two tableaux sharing a common border and moves them trough each other by decomposing the first tableau into a sequence
LINEAR TIME EQUIVALENCE OF LITTLEWOOD–RICHARDSON COEFFICIENT SYMMETRY MAPS
, 2009
"... Benkart, Sottile, and Stroomer have completely characterized by Knuth and dual Knuth equivalence a bijective proof of the Littlewood–Richardson coefficient conjugation symmetry, i.e. c λ µ,ν = c λt µ t,νt. Tableau–switching provides an algorithm to produce such a bijective proof. Fulton has shown th ..."
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Benkart, Sottile, and Stroomer have completely characterized by Knuth and dual Knuth equivalence a bijective proof of the Littlewood–Richardson coefficient conjugation symmetry, i.e. c λ µ,ν = c λt µ t,νt. Tableau–switching provides an algorithm to produce such a bijective proof. Fulton has shown that the White and the Hanlon–Sundaram maps are versions of that bijection. In this paper one exhibits explicitly the Yamanouchi word produced by that conjugation symmetry map which on its turn leads to a new and very natural version of the same map already considered independently. A consequence of this latter construction is that using notions of Relative Computational Complexity we are allowed to show that this conjugation symmetry map is linear time reducible to the Schützenberger involution and reciprocally. Thus the Benkart–Sottile–Stroomer conjugation symmetry map with the two mentioned versions, the three versions of the commutative symmetry map, and Schützenberger involution, are linear time reducible to each other. This answers a question posed by Pak and Vallejo.
Algebras
, 904
"... and groups defined by permutation relations of alternating type ∗ ..."
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Finitely presented algebras and groups defined by permutation relations
, 2008
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ACTIONS OF THE SYMMETRIC GROUP GENERATED BY COMPARABLE SETS OF INTEGERS AND SMITH INVARIANTS
 SÉMINAIRE LOTHARINGIEN DE COMBINATOIRE 59 (2010), ARTICLE B59G
, 2010
"... Lascoux and Schützenberger have shown that there exists a unique action of the symmetric group generated by the commutation of the column lengths of a twocolumn tableau and preserving the plactic class. We describe more general operators on pairs of comparable subsets of {1,..., n} which commute th ..."
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Lascoux and Schützenberger have shown that there exists a unique action of the symmetric group generated by the commutation of the column lengths of a twocolumn tableau and preserving the plactic class. We describe more general operators on pairs of comparable subsets of {1,..., n} which commute their cardinalities, and we prove that those operators define an action of the symmetric group by checking the braid relations on triples of sets of integers. The action of the symmetric group by Lascoux and Schützenberger appears in our construction as an extreme case as we only require the invariance of the shape and the weight of the insertion tableau. Instead of sets of positive integers one may take other equivalent objects as words in a two letter alphabet, and describe an action of the symmetric group on words congruent to keytableaux defined by reflection crystal operators type based on nonstandard pairing of parentheses. This construction arises naturally as a combinatorial description of the Smith invariants of certain sequences of products of matrices, over a local principal ideal domain, under a natural action of the symmetric group.
Key polynomials, invariant factors and an action of the symmetric group on Young tableaux
, 2007
"... We give a combinatorial description of the invariant factors associated with certain sequences of product of matrices, over a local principal ideal domain, under the action of the symmetric group by place permutation. Lascoux and Schützenberger have defined a permutation on a Young tableau to asso ..."
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We give a combinatorial description of the invariant factors associated with certain sequences of product of matrices, over a local principal ideal domain, under the action of the symmetric group by place permutation. Lascoux and Schützenberger have defined a permutation on a Young tableau to associate to each Knuth class a right and left key which they have used to give a combinatorial description of a key polynomial. The action of the symmetric group on the sequence of invariant factors generalizes this action of the symmetric group, by Lascoux and Schützenberger, to Young tableaux of the same shape and weight. As a dual translation, we obtain an action of the symmetric group on words congruent with keytableaux based on nonstandard pairing of parentheses.
A PATHTRANSFORMATION FOR RANDOM WALKS AND THE ROBINSONSCHENSTED CORRESPONDENCE
"... Abstract. The author and Marc Yor recently introduced a pathtransformation G (k) with the property that, for X belonging to a certain class of random walks on Z k +, the transformed walk G(k) (X) has the same law as the original walk conditioned never to exit the Weyl chamber {x: x1 ≤···≤xk}. In th ..."
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Abstract. The author and Marc Yor recently introduced a pathtransformation G (k) with the property that, for X belonging to a certain class of random walks on Z k +, the transformed walk G(k) (X) has the same law as the original walk conditioned never to exit the Weyl chamber {x: x1 ≤···≤xk}. In this paper, we show that G (k) is closely related to the RobinsonSchensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of X and G (k) (X). The corresponding results for the Brownian model are recovered by Donsker’s theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the pathtransformation G (k) and the RobinsonSchensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the RobinsonSchensted algorithm and, moreover, extends easily to a continuous setting. 1. Introduction and