GENERATORS FOR $\mathcal{H}$-INVARIANT PRIME IDEALS IN $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$

Abstract

It is known that, for generic q, the H-invariant prime ideals in Oq (Mm,p(C)) are generated by quantum minors (see S. Launois, Les idéaux premiers invariants de Oq (Mm,p(C)), J. Alg., in press). In this paper, m and p being given, we construct an algorithm which computes a generating set of quantum minors for each H-invariant prime ideal in Oq (Mm,p(C)). We also describe, in the general case, an explicit generating set of quantum minors for some particular H-invariant prime ideals in Oq (Mm,p(C)). In particular, if (Y i,α ) (i,α)∈[ [1,m] ]×[ [1,p] ] denotes the matrix of the canonical generators of Oq(Mm,p(C)),we prove that, if u 3, the ideal in Oq (Mm,p(C)) generated by Y 1,p and the u × u quantum minors is prime. This result allows Lenagan and Rigal to show that the quantum determinantal factor rings of Oq (Mm,p(C)) are maximal orders (see T. H. Lenagan and L. Rigal, Proc. Edinb. Math. Soc. 46 (2003), 513-529).Keywords: quantum matrices; quantum minors; prime ideals; quantum determinantal ideals; deleting-derivations algorithms 2000 Mathematics subject classification: Primary 16P40 Secondary 16W35; 20G42 IntroductionFix two positive integers m and p with m, p 2 and consider some complex number q which is transcendental over Q. Denote by R = O q (M m,p (C)) the quantization of the ring of regular functions on m × p matrices with entries in C (the field of complex numbers) and let (Y i,α (If m = p, this action is induced by the bialgebra structure of R and, if m = p, it is easy to check that the relations which define R are preserved by the group H.) It is known from work of Goodearl and Letzter that R has only finitely many Hinvariant prime ideals (see [8]) and that, in order to calculate the prime and primitive spectra of R, it is enough to determine the H-invariant prime ideals of R (see [8, Theorem 6.6]).In [10], we proved that the H-invariant prime ideals in R are generated by quantum minors, as conjectured by Goodearl and Lenagan (see [5] and [6]). In this paper, we use 163 164 S. Launois this result, together with Cauchon's description for the set of H-invariant prime ideals of R (see [3, Théorème 3.2.1]), to construct an algorithm which provides an explicit generating set of quantum minors for each H-invariant prime ideal in R (see § 4). (Of course, these generating sets can be computed with this algorithm only when m and p have fixed values.)The last part of this paper is devoted to the general case. We construct certain sets of quantum minors which generate prime ideals of R. In order to do that, we consider a new deleting-derivations algorithm (see [2]) that we define in § 5. Using this new tool, we can prove that, if u 3, the ideal in O q (M m,p (C)) generated by Y 1,p and the u × u quantum minors is prime. This result allows Lenagan and Rigal [11] to show that the quantum determinantal factor rings of O q (M m,p (C)) are maximal orders. H-invariant prime ideals in O q (M m,p (C))Throughout this paper, we use the following conventions.(i) N, Q and C denote, respectively, the set of natur...

Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

“…Hence, δ ( j,β) and δ ( j,α h ) α h →β t j,α h both belong to L. Finally, we note that [17,Theorem 3.7], which was proved for the case K = C, holds for the general coefficient field K (with the same proof). This result implies that the powers of u = t j,β are linearly independent over L. It follows that δ ( j,β) = 0, as desired.…”

“…Although in [17] an algorithm was developed that constructs, starting only from a Cauchon diagram C, all of the quantum minors that belong to the H-prime ideal J C , it is not easy to identify the families of quantum minors that generate H-prime ideals. Casteels [3] has recently developed a graph theoretic method in order to compute these families.…”

“…The proof of this result (see Sect. 4) relies on two algorithms, the deleting-derivations algorithm and its inverse the restoration algorithm, that were first developed for use in quantum matrices [5,6,17]. We note that recently and independently Casteels [3] has developed graph-theoretic methods (also based on the restoration algorithm) to compute the set of quantum minors that belong to a torus-invariant prime in…”

The Maximal Order Property for Quantum Determinantal Rings

“…The problem is a technical one: it is necessary to show that a certain ideal is a prime ideal. The restriction to q an element of C transcendental over Q is because the relevant ideal is shown to be prime in [6] for this case. We can answer the question for general 0 = q ∈ K for the case of R n (X) (= O q (M n )/ det q ), when X is n × n, and also for the case R 3 (X) for general m × n. A proof of the former case is included in this paper since it is relatively short, and of independent interest.…”

Quantum matrices by paths

“…(3) There is an injective homomorphism ← − · : [15], and a step of the "restoration" algorithm in [9]. )…”