Plactic Algebra Research Articles

A Plactic Algebra Action on Bosonic Particle Configurations

We study an action of the plactic algebra on bosonic particle configurations. These particle configurations together with the action of the plactic generators can be identified with crystals of the quantum analogue of the symmetric tensor representations of the special linear Lie algebra mathfrak {s} mathfrak {l}_{N}. It turns out that this action factors through a quotient algebra that we call partic algebra, whose induced action on bosonic particle configurations is faithful. We describe a basis of the partic algebra explicitly in terms of a normal form for monomials, and we compute the center of the partic algebra.

Tropical plactic algebra, the cloaktic monoid, and semigroup representations

A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetic, encapsulating the ubiquitous plactic monoid Pn. This algebra manifests a natural framework for accommodating representations of Pn, and equivalently of Young tableaux, and its moderate coarsening — the cloaktic monoid Kn and the co-cloaktic monoid ▪. The faithful linear representations of Kn and ▪ by tropical matrices, which constitute a tropical plactic algebra, are shown to provide linear representations of the plactic monoid. To this end the paper develops a special type of configuration tableaux, corresponding bijectively to semi-standard Young tableaux. These special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways, including algorithmic benefits. The interplay between these algebraic-combinatorial structures establishes a profound machinery for exploring semigroup attributes, in particular satisfying of semigroup identities. This machinery is utilized here to prove that Kn and ▪ admit all the semigroup identities satisfied by n×n triangular tropical matrices, which holds also for P3.

Irreducible representations of the plactic algebra of rank four

Irreducible representations of the plactic monoid M of rank four are studied. Certain concrete families of simple modules over the plactic algebra K[M] over a field K are constructed. It is shown that the Jacobson radical J(K[M]) of K[M] is nilpotent. Moreover, the congruence ρ on M determined by J(K[M]) coincides with the intersection of the congruences determined by the primitive ideals of K[M] corresponding to the constructed simple modules. In particular, M/ρ is a subdirect product of the images of M in the corresponding endomorphism algebras.

Cohomology rings of the plactic monoid algebra via a Gröbner–Shirshov basis

In this paper, we calculate the cohomology ring [Formula: see text] and the Hochschild cohomology ring of the plactic monoid algebra [Formula: see text] via the Anick resolution using a Gröbner–Shirshov basis.

Notes on Schubert, Grothendieck and Key Polynomials

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.

New approaches to plactic monoid via Gröbner–Shirshov bases

We give two explicit (quadratic) presentations of the plactic monoid in row and column generators correspondingly. Then we give direct independent proofs that these presentations are Gröbner–Shirshov bases of the plactic algebra in deg-lex orderings of generators. From Composition-Diamond lemma for associative algebras it follows that the set of Young tableaux is the Knuth normal form for plactic monoid ([30], see also Ch. 5 in [32]).

Finite Gröbner–Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids

This paper shows that every Plactic algebra of finite rank admits a finite Gröbner–Shirshov basis. The result is proved by using the combinatorial properties of Young tableaux to construct a finite complete rewriting system for the corresponding Plactic monoid, which also yields the corollaries that Plactic monoids of finite rank have finite derivation type and satisfy the homological finiteness properties left and right FP∞. Also, answering a question of Zelmanov, we apply this rewriting system and other techniques to show that Plactic monoids of finite rank are biautomatic.

Gröbner-Shirshov Bases for Plactic Algebras

A finite Gröbner-Shirshov basis is constructed for the plactic algebra of rank 3 over a field K. It is also shown that plactic algebras of rank exceeding 3 do not have finite Gröbner-Shirshov bases associated to the natural degree-lexicographic ordering on the corresponding free algebra. The latter is in contrast with the case of a strongly related class of algebras, called Chinese algebras.

Plactic algebra of rank 3

The structure of the algebra K[M] of the plactic monoid M of rank 3 over a field K is studied. The minimal prime ideals of K[M] are described. There are only two such ideals and each of them is a principal ideal determined by a homogeneous congruence on M. Moreover, in case K is uncountable and algebraically closed, the left and right primitive spectrum and the corresponding irreducible representations of the algebra K[M] are described. All these representations are monomial. As an application, a new proof of the semiprimitivity of K[M] is given.

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

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.

Noncommutative Schur polynomials and the crystal limit of the -vertex model

Starting from the Verma module of we consider the evaluation module for affine and discuss its crystal limit (q → 0). There exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations. Its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra, an extension of the finite plactic algebra first discussed by Lascoux and Schützenberger. The corresponding noncommutative elementary symmetric polynomials were recently shown to be generated by the transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin and Kitanine. Here we establish that both generating functions satisfy Baxter's TQ-equation in the crystal limit by tying them to special solutions of the Yang–Baxter equation. The TQ-equation amounts to the well-known Jacobi–Trudi formula leading naturally to the definition of noncommutative Schur polynomials. The latter can be employed to define a ring which has applications in conformal field theory and enumerative geometry: it is isomorphic to the fusion ring of the Wess–Zumino–Novikov–Witten model whose structure constants are the dimensions of spaces of generalized θ-functions over the Riemann sphere with three punctures.

Crystal bases and combinatorics of infinite rank quantum groups

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.

Bitableaux bases of the quantum coordinate algebra of a semisimple group

We extend the Standard Basis Theorem of Rota et al. to the setting of quantum symmetrizable Kac–Moody algebras. In particular, we obtain a procedure to give a presentation of the quantum coordinate algebra of any semisimple group, for generic q. More precisely, given any integrable module V of a quantum symmetrizable Kac–Moody algebra U q ( g ) , we obtain a generating set of the ideal of relations among the matrix coefficients of V, and we give an upper bound for the degrees of these polynomials. Our approach is based on the theory of crystal bases and Littelmann's generalization of the plactic algebra.

Twisting of homogeneous algebras

Twisting process for homogeneous algebras is defined. It consists of a particular kind of actual deformations controlled by a multiplicative cosimplicial quasicomplex C • related to each homogeneous algebra. The elements implementing twist transformations are a subclass of the counital 2-cocycles of C • . Such transformations are studied by means of their co-homological properties. It is shown they define an equivalence relation between homogeneous algebras, in such a way that cubic Artin–Schelter regular algebras of type S 1 (including parafermionic and plactic algebras with two generators) belong to the same class.

Plactic algebras

The structure of the plactic algebra K[ M] of rank 2 or 3 is studied. It is shown that these algebras are semiprimitive. Moreover, the prime spectrum is completely described in the rank 2 case. Also it is proved that the plactic algebra of rank n⩾4 is not semiprime and the plactic algebra of rank 3 is not prime.

Homogeneous Algebras, Statistics and Combinatorics

After some generalities on homogeneous algebras, we give a formula connecting the Poincare series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: The first one, called the parafermionic (parabosonic) algebra, is the algebra generated by the creation operators of the universal fermionic (bosonic) parastatics with D degrees of freedom while the second is the plactic algebra, i.e., the algebra of the plactic monoid with entries in {1, 2,..., D}. In the case D=2, we describe the relations with the cubic Artin–Schelter algebras. It is pointed out that the natural action of GL(2) on the parafermionic algebra for D=2 extends as an action of the quantum group GLp,q(2) on the generic cubic Artin–Schelter regular algebra of type S1; p and q being related to the Artin–Schelter parameters. It is claimed that this has a counterpart for any integer D⩾2.

A Plactic Algebra for Semisimple Lie Algebras

A plactic algebra can be thought of as a (non-commutative) model for the representation ring of a semisimple Lie algebra g. This algebra was introduced by Lascoux and Schutzenberger in [13], [18] in order to study the representation theory of GLn(C) and Sn. This new tool enabled them for example to give the first rigouros proof of the Littlewood-Richardson rule to determine the decomposition of tensor products into direct sums of irreducible representations. Using a case by case analysis, such a plactic algebra has been constructed also for some other simple groups, see [1], [8], [19], [20], [21]. Recently, two constructions of isomorphic plactic algebras have been given for symmetrisable Kac-Moody algebras. From the point of view of quantum groups, this algebra is the algebra of crystal bases ([5], [6], [7], [16], [17], [19]). The second construction realizes this algebra as the algebra ZP of equivalence classes of paths in the space XQ of rational weights ([5], [14], [15]). For simplicity, assume that G is a simple, simply connected algebraic group. To give a description of ZP which is more in the spirit of the original work of Lascoux and Schutzenberger, let V = Vλ1 ⊕ . . .⊕Vλr be a faithful representation and let D be the associated set of L-S paths, i.e. D is a basis of the corresponding model of V in ZP. Let Z{D} be the free associative algebra generated by D. If λ = ∑ aωω is a dominant weight, then let |λ| denote the sum ∑ aω. The canonical projection which maps a monomial to the concatenation:

THE RULE OF LITTLEWOOD-RICHARDSON IN A CONSTRUCTION OF BERENSTEIN-ZELEVINSKY

The rule of Littlewood-Richardson gives the decomposition of a product of Schur functions in the basis of the same functions. Each coefficient of this decomposition is the number of factorizations of a tableau of Yamanouchi in the plactic algebra. A. D. Berenstein and A. V. Zelevinksy prove that these coefficients are also the numbers of certain configurations called triangles. This text gives an explicit bijection between these triangles and the words of Yamanouchi.