Hopf dreams and diagonal harmonics

Abstract

This paper introduces a Hopf algebra structure on a family of reduced pipe dreams. We show that this Hopf algebra is free and cofree, and construct a surjection onto a commutative Hopf algebra of permutations. The pipe dream Hopf algebra contains Hopf subalgebras with interesting sets of generators and Hilbert series related to subsequences of Catalan numbers. Three other relevant Hopf subalgebras include the Loday-Ronco Hopf algebra on complete binary trees, a Hopf algebra related to a special family of lattice walks on the quarter plane, and a Hopf algebra on 𝜈-trees related to 𝜈-Tamari lattices. One of this Hopf subalgebras motivates a new notion of Hopf chains in the Tamari lattice, which are used to present applications and conjectures in the theory of multivariate diagonal harmonics. M S C ( 2 0 2 0 ) 16T05, 16T30, 05E10 (primary) Contents

Parabolic Tamari Lattices in Linear Type B

“…As [ [1]] is a simple reflection and the only decomposition ((1 3)) = ((1 2)) + ((2 3)) must be discarded because ((1 2)) / ∈ Inv(ω o;α ), we conclude that w is w o;α ( c)-aligned. However, the inversion t = [ [3]] can be decomposed as…”

“…Besides the structural considerations, it may also be fruitful to consider various enumerative aspects of type-B parabolic Tamari lattices. For instance, the total sum of the cardinalities of parabolic aligned elements with respect to all parabolic quotients of a fixed symmetric group yields the sequence [30, A151498] and enumerates the dimensions of a certain family of Hopf algebras, see [1,6]. The counterpart in type B is the sequence t 1 , t 2 , .…”

“…] denote the sign-symmetric permutation that exchanges i and −i, and ((i j)) the sign-symmetric permutation that exchanges the values i and j (and simultaneously −i and −j), where we do not make assumptions about the sign of i and j. Then, the assignment s 0 → [ [1]] and s i → ((i i+1)) for i ∈ [n − 1] allows us to switch from the Coxeter representation of H n to its permutation representation (when multiplying by s i on the left of a sign-symmetric permutation). Via the induced isomorphism, we may transfer the notions of reflection set, Coxeter length, (cover) inversions, and weak order to sign-symmetric permutations.…”

“…and neither of these decompositions can be discarded, because [ [1]], [ [2]], ((2 3)), ((1 3)) ⊆ Inv(ω o;α ). As we have [ [2]] ≺ ((2 3)) and [ [2]] / ∈ Inv(w), the condition from Definition 15 would fail for w if all inversions were considered.…”

$(GL_k\times S_n)$-Modules of Multivariate Diagonal Harmonics

“…In parallel, an explicit study of E 3 n (obtained by closing E 2 n with respect to polarization operators involving a third set of variables) was started about 10 years ago. Most fundamental questions about it remain open, but its study has suggested new (hard to prove) combinatorial identities linked to the Tamari Lattice as discussed in [11,12]. The general k-framework is also considered in a previous paper [7], where some broad properties of the modules of k-variate diagonal coinvariant (for any finite complex reflection groups), considering the inductive limit E n := lim k→∞ E k n as a GL ∞ ×S n -module (with commuting actions), and its decomposition:…”