Narrow Results

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.

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.

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.

In this paper, we calculate the cohomology ring ${\mathrm{\text{Ext}}}_{\mathrm{\text{𝕜}}{\mathrm{\text{Pl}}}_{\mathrm{\text{n}}}}^{\ast }(\mathrm{\text{𝕜}},\mathrm{\text{𝕜}})$ and the Hochschild cohomology ring of the plactic monoid algebra $\mathrm{\text{𝕜}}{\mathrm{\text{Pl}}}_n$ via the Anick resolution using a Gröbner–Shirshov basis.

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.

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.

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.

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.