RANKtℋ-PRIMES IN QUANTUM MATRICES

Abstract

For q ∈ C generic, we give an algorithmic construction of an ordered bijection between the set of H-primes of O q (M n (C)) and the sub-poset S of the (reverse) Bruhat order of the symmetric group S 2n consisting of those permutations that move any integer by no more than n positions. Further, we describe the permutations that correspond via this bijection to rank t H-primes. More precisely, we establish the following result. Imagine that there is a barrier between positions n and n + 1. Then a 2n-permutation σ ∈ S corresponds to a rank t H-invariant prime ideal of O q (M n (C)) if and only if the number of integers that are moved by σ from the right to the left of this barrier is exactly n − t.The existence of such a bijection was conjectured by Goodearl and Lenagan.2000 Mathematics subject classification: 16W35, 20G42, 06A07.Proposition 2.3 The map χ : S n × S n → H ′ R -Spec(O q (SL n (C))) defined by χ(w) = I w is an order-reversing bijection; its inverse is also an order-reversing bijection.Proof. In view of Proposition 2.2, it just remains to show that χ and χ −1 are decreasing. Concerning χ, it easily follows from the characterization of the (reverse) Bruhat ordering given in Proposition 1.2 and the definition of I w . So it just remains to deal with χ −1 . Let J ⊆ K be two H ′ R -invariant prime ideals of O q (SL n (C)). There exist w J and w K in S n × S n such that J = I w J and K = I w K . In order to prove that χ −1 is decreasing, we need to show thatAssume that this is not the case. Then we have for instance w + J w + K . It follows from Proposition 1.2 that there exists j ∈ [[1, n − 1]] such that w + J j w + K and so we have c + j,w + K ∈ I w J = J ⊆ K. Hence the quantum minor c + j,w + K must belong to K = I w K . But, on the other hand, Hodges and Levasseur have shown (see [11, Theorem 2.2.3]) that, for all i ∈ [[1, n − 1]], c + i,w + K ∈ I w K . This is a contradiction and so we have w + J ≥ w + K and w − J ≥ w − K , as required.

Admissible diagrams in quantum nilpotent algebras and combinatoric properties of Weyl groups

“…Let us recall that, in the particular case U + [w] = O q (M p,m (k)) mentioned above, S. Launois [14] has constructed (with quite different methods) such a one-to-one correspondence which, moreover, preserves the ordering (where the Weyl group is provided with the Bruhat order and the spectrum Spec(O q (M p,m (k))) is provided with the inclusion order). This leads us to ask the following questions (unsolved at the moment):…”

Dimension and enumeration of primitive ideals in quantum algebras

“…More precisely, we show that the number of primitive H-invariant prime ideals in O q (M 2,n ) is (3 n+1 − 2 n+1 + (−1) n+1 + 2)/4. Cauchon [5] (see also [14]) enumerated the Hinvariant prime ideals in O q (M n ), giving a closed formula in terms of the Stirling numbers of the second kind. In particular, the number of H-invariant prime ideals in O q (M 2,n ) is 2 · 3 n − 2 n .…”

A Formula for Multi-Poly-Bernoulli Numbers of Negative Index

“…Moreover, these numbers have combinatorial applications; see [2] and [7] for details. [4] gave some recurrence relations for multi-poly-Bernoulli numbers, and gave a duality formula for specialized multipoly-Bernoulli numbers [3] (see also Corollary 3(ii) below).…”