Hopf algebra structure on packed square matrices

Abstract

International audienceWe construct a new bigraded Hopf algebra whose bases are indexed by square matrices with entries in the alphabet {0, 1,. .. , k}, k 1, without null rows or columns. This Hopf algebra generalizes the one of permutations of Malvenuto and Reutenauer, the one of k-colored permutations of Novelli and Thibon, and the one of uniform block permutations of Aguiar and Orellana. We study the algebraic structure of our Hopf algebra and show, by exhibiting multiplicative bases, that it is free. We moreover show that it is self-dual and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra, we obtain a new one indexed by alternating sign matrices. We study some of its properties and algebraic quotients defined through alternating sign matrices statistics

Weak order and descents for monotone triangles

“…As described in the Introduction, the map w → Des(w) that sends a permutation w in S n to its descent set was pleasingly reinterpreted in the work of Malvenuto and Reutenauer [11] as a morphism of Hopf algebras. We wish to explain here how this extends to the map T → Des(T ) sending a monotone triangle to its descent set, giving at least an algebra (but not coalgebra) morphism out of the Hopf algebra of ASM s recently defined by Cheballah, Giraudo and Maurice [8].…”

“…Here L α denotes Gessel's fundamental quasisymmetric function associated to a composition α, and α(Des(w)) is the composition whose partial sums give the elements of Des(w); see [16, §7.19] and Section 7 below. Recently, Cheballah, Giraudo and Maurice embedded FQSym inside a larger graded Hopf algebra ASM whose n th -graded component has a basis {A} indexed by A in ASM n [8], and whose product and coproduct extend that of FQSym. Section 7 proves the following.…”

“…, so that if we construct T ′ from T by replacing T k with H max (I, J) as defined in (8), then it will certainly have…”

“…, s 8 } = S, then the parabolic subgroup J inside S 9 is the subgroup isomorphic to S 3 × S 4 × S 2 that stabilizes the blocks of the partition {1, 2, 3}, {4, 5, 6, 7}, {8, 9}. Its longest permutation is w 0 (J) =(3,2,1,7,6,5,4,9,8).…”

“…Actually, in[8] the algebra structure uses column shuffles, but this is equivalent to what is described here after transposing the alternating sign matrices A → A t .…”