arXiv subjects
Publications and source records attributed to T H Lenagan.
We establish a factorisation theorem for invertible, cross-symmetric, totally nonnegative matrices, and illustrate the theory by verifying that certain cases of Holte's Amazing Matrix are totally nonnegative.
Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.
Tauvel's height formula, which provides a link between the height of a prime ideal and the Gelfand-Kirillov dimension of the corresponding factor algebra, is verified for quantum nilpotent algebras.
A uniqueness theorem for an LU decomposition of a totally nonnegative matrix is obtained.
It is shown that over an arbitrary countable field, there exists a finitely generated algebra that is nil, infinite dimensional, and has Gelfand-Kirillov dimension at most three.
Explicit generating sets are found for all primitive ideals in the generic quantized coordinate rings of the 3x3 special and general linear groups over an arbitrary algebraically closed field. (Previously, generators were only known up to certain localizations.) The generating sets form polynormal regular sequences, from which it follows that all primitive factor algebras of these quantized coordinate rings are Auslander-Gorenstein and Cohen-Macaulay.
In contrast to the classical and semiclassical settings, the Coxeter element (12...n) which cycles the columns of an mxn matrix does not determine an automorphism of the quantum grassmannian. Here, we show that this cycling can be obtained by defining a cocycle twist. A consequence is that the torus invariant prime ideals of the quantum grassmannian are permuted by the action of the Coxeter element (12...n); we view this as a quantum analogue of the recent result of Knutson, Lam and Speyer that the Lusztig strata of the classical grassmannian are permuted by (12...n).
The quantum grassmannian is known to be a graded quantum algebra with a straightening law when the poset of generating quantum minors is endowed with the standard partial ordering. In this paper it is shown that this result remains true when the ordering is subjected to cyclic shifts. The method involves proving that noncommutative dehomogenisation is possible at any consecutive quantum minor.
We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a torus H that acts naturally on the quantum grassmannian and the cell decomposition of the set of H-primes leads to a parameterisation of the H-spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the non-negative cells in recent studies concerning the totally non-negative grassmannian. Finally, we use the cell decomposition to establish that the quantum grassmannian satisfies normal separation and catenarity.
We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains which are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.
We calculate the first Hochschild cohomology group of quantum matrices, the quantum general linear group and the quantum special linear group in the generic case when the deformation parameter is not a root of unity. As a corollary, we obtain information about twisted Hochschild homology of these algebras.
Let $K$ be an arbitrary field of characteristic zero, $P_n:= K[ x_1, ..., x_n]$ be a polynomial algebra, and $P_{n, x_1}:= K[x_1^{-1}, x_1, ..., x_n]$, for $n\geq 2$. Let $\s' \in {\rm Aut}_K(P_n)$ be given by $$ x_1\mapsto x_1-1, \quad x_1\mapsto x_2+x_1,\quad ... ,\quad x_n\mapsto x_n+x_{n-1}.$$ It is proved that the algebra of invariants, $F_n':= P_n^{\s'}$, is a polynomial algebra in $n-1$ variables which is generated by $[\frac{n}{2}]$ quadratic and $[\frac{n-1}{2}]$ cubic (free) generators that are given explicitly. Let $\s \in {\rm Aut}_K(P_n)$ be given by %$$\s \in {\rm Aut}_K(P_n): $$ x_1\mapsto x_1, \quad x_1\mapsto x_2+x_1, \quad ... ,\quad x_n\mapsto x_n+x_{n-1}.$$ It is well-known that the algebra of invariants, $F_n:= P_n^\s$, is finitely generated (Theorem of Weitzenböck, \cite{Weitz}, 1932), has transcendence degree $n-1$, and that one can give an explicit transcendence basis in which the elements have degrees $1, 2, 3, ..., n-1$. However, it is an old open problem to find explicit generators for $F_n$. We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants $P_{n, x_1}^\s$ is a polynomial algebra over $K[x_1, x_1^{-1}]$ in $n-2$ variables which is generated by $[\frac{n-1}{2}]$ quadratic and $[\frac{n-2}{2}]$ cubic (free) generators that are given explicitly. The coefficients of these quadratic and cubic invariants throw light on the `unpredictable combinatorics' of invariants of affine automorphisms and of $SL_2$-invariants.
We show that the quantum coordinate ring of a semisimple group is a unique factorisation domain in the sense of Chatters and Jordan in the case where the deformation parameter q is a transcendental element.
Let q be a nonzero complex number that is not a root of unity. We give a criterion for (0) to be a primitive ideal of the algebra O_q(M_{m,n}) of quantum matrices. Next, we describe all height one primes of O_q(M_{m,n}); these two problems are actually interlinked since it turns out that (0) is a primitive ideal of O_q(M_{m,n}) whenever O_q(M_{m,n}) has only finitely many height one primes. Finally, we compute the automorphism group of O_q(M_{m,n}) in the case where m is not equal to n. In order to do this, we first study the action of this group on the prime spectrum of O_q(M_{m,n}). Then, by using the preferred basis of O_q(M_{m,n}) and PBW bases, we prove that the automorphism group of O_q(M_{m,n}) is isomorphic to the torus (C*)^{m+n-1} when m is not equal to n, and (m,n) is not equal to (1,3) and (3,1).
We construct a nil algebra over a countable field which has finite but non-zero Gelfand-Kirillov dimension.
We prove a general theorem showing that iterated skew polynomial extensions of the type which fit the conditions needed by Cauchon's deleting derivations theory and by the Goodearl-Letzter stratification theory are unique factorisation rings in the sense of Chatters and Jordan. This general result applies to many quantum algebras; in particular, generic quantum matrices and quantized enveloping algebras of the nilpotent part of a semisimple Lie algebra are unique factorisation domains in the sense of Chatters. By using noncommutative dehomogenisation, the result also extends to generic quantum grassmannians.
We study quantum analogues of quotient varieties, namely quantum grassmannians and quantum determinantal rings, from the point of view of regularity conditions. More precisely, we show that these rings are AS-Cohen-Macaulay and determine which of them are AS-Gorenstein. Our method is inspired by the one developed by De Concini, Eisenbud and Procesi in the commutative case. Thus, we introduce and study the notion of a quantum graded algebra with a staightening law on a partially ordered set, showing in particular that, among such algebras, those whose underlying poset is wonderful are AS-Cohen-Macaulay. Then, we prove that both quantum grassmannians and quantum determinantal rings are quantum graded algebras with a staightening law on a wonderful poset, hence showing that they are AS-Cohen-Macaulay. In this last step, we are lead to introduce and study (to some extent) natural quantum analogues of Schubert varieties.
The traces of the quantum powers of a generic quantum matrix pairwise commute. This was conjectured by Kaoru Ikeda, in connection with certain Hamiltonian systems. The proof involves Newton's formulae for quantum matrices, relating traces of quantum powers with sums of principal minors.