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  • articleNo Access

    SUPER RSK-ALGORITHMS AND SUPER PLACTIC MONOID

    We construct the analog of the plactic monoid for the super semistandard Young tableaux over a signed alphabet. This is done by developing a generalization of the Knuth's relations. Moreover we get generalizations of Greene's invariants and Young–Pieri rule. A generalization of the symmetry theorem in the signed case is also obtained. Except for this last result, all the other results are proved without restrictions on the orderings of the alphabets.

  • articleNo Access

    Finite convergent presentation of plactic monoid for type C

    We give an explicit presentation for the plactic monoid for type C using admissible column generators. Thanks to the combinatorial properties of symplectic tableaux, we prove that this presentation is finite and convergent. We obtain as a corollary that plactic monoids for type C satisfy homological finiteness properties.

  • articleNo Access

    Quadratic normalization in monoids

    In the general context of presentations of monoids, we study normalization process that are determined by their restriction to length-two words. Garside’s greedy normal forms and quadratic convergent rewriting systems, in particular those associated with the plactic monoids, are typical examples. Having introduced a parameter, called the class and measuring the complexity of the normalization of length-three words, we analyze the normalization of longer words and describe a number of possible behaviors. We fully axiomatize normalizations of class (4,3), show the convergence of the associated rewriting systems, and characterize those deriving from a Garside family.

  • articleNo Access

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

    In this paper, we calculate the cohomology ring Ext𝕜Pln(𝕜,𝕜) and the Hochschild cohomology ring of the plactic monoid algebra 𝕜Pln via the Anick resolution using a Gröbner–Shirshov basis.

  • articleNo Access

    Division in the plactic monoid

    In [D. R. L. Brown, Plactic key agreement (insecure?), J. Math. Cryptol.17(1) (2023) 20220010], a novel cryptographic key exchange technique was proposed using the plactic monoid, based on the apparent difficulty of solving division problems in that monoid. Specifically, given elements c,b in the plactic monoid, the problem is to find q for which qb=c, given that such a q exists. In this paper, we introduce a metric on the plactic monoid and use it to give a probabilistic algorithm for solving that problem which is fast for parameter values in the range of interest.

  • articleNo Access

    PARASTATISTICS ALGEBRA, YOUNG TABLEAUX AND THE SUPER PLACTIC MONOID

    The parastatistics algebra is a superalgebra with (even) parafermi and (odd) parabose creation and annihilation operators. The states in the parastatistics Fock-like space are shown to be in one-to-one correspondence with the Super Semistandard Young Tableaux (SSYT) subject to further constraints. The deformation of the parastatistics algebra gives rise to a monoidal structure on the SSYT which is a super-counterpart of the plactic monoid.

  • articleNo Access

    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.

  • chapterNo Access

    Some New Results on Gröbner-Shirshov Bases

    In this survey article, we report some new results of Gröbner-Shirshov bases, including new Composition-Diamond lemmas and some applications of some known Composition-Diamond lemmas.