相关论文: Schensted-type correspondences and plactic monoids…

Let $\mathsf{mSt}_n$ be the plactic-like monoid obtained by factoring the free monoid over a finite alphabet $\mathcal{A}_n$ by the meet of the stalactic congruence and its dual. In this paper, we prove that $\mathsf{mSt}_n$ can be equipped…

We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a "nice" Garside element, certain monoids $S$ with quadratic relations, whose monoidal…

The hypoplactic monoid was introduced by Krob and Thibon through a presentation and through quasi-ribbon tableaux and an insertion algorithm. Just as Kashiwara crystals enriched the structure of the plactic monoid and allowed its…

Serrano (2010) introduced the shifted plactic monoid, governing Haiman's (1989) mixed insertion algorithm, as a type B analogue of the classical plactic monoid that connects jeu de taquin of Young tableaux with the Robinson-Schensted-Knuth…

Young tableaux carry an associative product, described by the Schensted algorithm. They thus form a monoid $\mathbf{Pl}$, called \emph{plactic}. It is central in numerous combinatorial and algebraic applications. In this paper, the tableaux…

We provide an explicit combinatorial description of the embedding of the crystal of Kashiwara-Nakashima tableaux in types $B$ and $C$ into that of $\bi$-Lusztig data associated to a family of reduced expressions $\bi$ of the longest element…

We study skein modules using $Sp(2n)$ webs. We define multivariable analogues of Chebyshev polynomials in the Type $C$ setting and use them to construct transparent elements in the skein module at roots of unity. Our arguments are…

The surface singular braid monoid corresponds to marked graph diagrams of knotted surfaces in braid form. In a quest to resolve linearity problem for this monoid, we will show that if it is defined on at least two or at least three strands,…

The theory of Newton-Okounkov polytopes is a generalization of that of Newton polytopes for toric varieties, and it gives a systematic method of constructing toric degenerations of a projective variety. In the case of Schubert varieties,…

We extend a quantized skew Howe duality result for Type $\mathbf{A}$ algebras to orthogonal types via a seesaw. We develop an operator commutant version of the First Fundamental Theorem of invariant theory for $U_q(\mathfrak{so}_n)$ using a…

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…

Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q(\mathfrak{sl}_{n+1})$, $U_q(\mathfrak{sp}_{2n})$ and…

In the recent papers with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type $B$. Namely, we conjectured that…

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and…

We introduce a monoid corresponding to knotted surfaces in four space, from its hyperbolic splitting represented by marked diagram in braid like form. It has four types of generators: two standard braid generators and two of singular type.…

Crystal graphs are powerful combinatorial tools for working with the plactic monoid and symmetric functions. Quasi-crystal graphs are an analogous concept for the hypoplactic monoid and quasi-symmetric functions. This paper makes a…

We construct finite coherent presentations of plactic monoids of type A. Such coherent presentations express a system of generators and relations for the monoid extended in a coherent way to give a family of generators of the relations…

We study the representation theory of a quantum symmetric pair $(\mathbf{U},\mathbf{U}^{\jmath})$ with two parameters $p,q$ of type AIII, by using highest weight theory and a variant of Kashiwara's crystal basis theory. Namely, we classify…

The Sch\"utzenberger product of monoids is a key tool for the algebraic treatment of language concatenation. In this paper we generalize the Sch\"utzenberger product to the level of monoids in an algebraic category $\mathscr{D}$, leading to…

The free monoid $A^*$ on a finite totally ordered alphabet $A$ acts at the left on columns, by Schensted left insertion. This defines a finite monoid, denoted $Styl(A)$ and called the stylic monoid. It is canonically a quotient of the…