𝑎𝑑-nilpotent 𝔟-ideals in 𝔰𝔩(𝔫) having a fixed class of nilpotence: combinatorics and enumeration

Abstract

Abstract. We study the combinatorics of ad-nilpotent ideals of a Borel subalgebra of sl(n + 1, C). We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between ad-nilpotent ideals and Dyck paths. Finally, we propose a (q, t)-analogue of the Catalan number Cn. These (q, t)-Catalan numbers count, on the one hand, ad-nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.

A transitivity result for ad-nilpotent ideals in type A

“…Since λ(O I ) = λ I by Corollary 6.2, it is clear that (λ I ) 1 is the smallest positive integer k such A k = 0 for all A ∈ I. Moreover, it is pointed out in [11, p. 536] that (λ I ) 1 is the index of nilpotence of I as an associative algebra, while in [2] it is shown that (λ I ) 1 is the index of nilpotence of I as a Lie algebra (actually, the latter use the class of nilpotence to refer to the number of brackets needed to get to zero, which equals (λ I ) 1 − 1, i.e., the bounce count). We will say that I has index k.…”

“…. , i r ) of I was subsequently re-introduced in the literature [2] and then adapted to Dyck paths [12], where it determines the bounce path. Visually, this is the lattice path that traces the lower boundary of the ideal I := n P where P = P i2−1,...,ir−1 .…”

“…Enumeration. Several results related to enumerating the ideals of index k, hence also those of corank k, are given in [2]. It would be nice to be able to find a formula for the cardinality N λ of the equivalence class attached to O λ .…”

An area-depth symmetric $q,t$-Catalan polynomial

“…This proves that (2.3) and (2.5) are equal. The inverse of the zeta map first appeared connection with nilpotent ideals in certain Borel subalgebras of sl(n) [AKOP02]. For its connections with the combinatorics of q, t-Catalan polynomials, see [Hag08].…”

“…Shortly thereafter, Haiman announced a different combinatorial formula using the area and dinv statistics (see (2.3)). The zeta map [AKOP02,Hag08] relates these two combinatorial formulas. One of the main open problems related to the q, t-Catalan polynomials C n (q, t) is a combinatorial proof of its symmetry in q and t.…”