Quantum Matrices by paths

Quantum Matrices by paths

Karel Casteels School of Mathematics, Statistics and Actuarial Science
University of Kent
Canterbury, United Kingdom

We study, from a combinatorial viewpoint, the quantized coordinate ring of matrices over an infinite field , (often simply called quantum matrices).The first part of this paper shows that , which is traditionally defined by generators and relations, can be seen as a subalgebra of a quantum torus by using paths in a certain directed graph. Roughly speaking, we view each generator of as a sum over paths in the graph, each path being assigned an element of the quantum torus. The relations then arise naturally by considering intersecting paths. This viewpoint is closely related to Cauchon’s deleting derivations algorithm.

The second part of this paper applies the above to the theory of torus-invariant prime ideals of . We prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum parameter is a non-root of unity, have generating sets consisting of quantum minors. Previously, this result was known to hold only when and with transcendental over . Our strategy is to prove the stronger result that the quantum minors in a given torus-invariant ideal form a Gröbner basis.

1. Introduction

The purpose of this paper is to introduce a “combinatorial model” of , the quantized coordinate ring of matrices over a field (simply called quantum matrices). We demonstrate the utility of this model by using it to study the prime spectrum of .

Quantum matrices have generated a good deal of interest since their discovery during the initial development of quantum group theory in the 1980’s. This is because not only do quantum matrices underlie many of the traditional quantum groups such as the quantum special and general linear groups, but there are also interesting connections with topics such as braided tensor categories and knot theory. See [20] for a brief survey. More recently, it has been observed [8, 9, 17] that the prime spectrum of quantum matrices is deeply related to the theory of totally nonnegative matrices and the totally nonnegative grassmannian in the sense of Postnikov [19].

Since the late 1990’s, much effort has been expended toward understanding the structure of the prime and primitive spectra of various quantum algebras. Quantum matrices have received particular attention since, while this algebra has a seemingly simple structure (for example, it is an iterated Ore extension over the field ), many problems have proven difficult to resolve. In particular, the machineries employed to analyze have tended to use fairly sophisticated viewpoints from noncommutative ring theory and representation theory and even then often require extra restrictions on the base field and choice of quantum parameter .

The -stratification theory of Goodearl and Letzter [13] (see also  [2]) is an important advancement toward understanding the prime and primitive spectra of some quantum algebras. Briefly, many noncommutative rings support a rational action of a torus which allows one to partition the prime spectrum of the ring into finitely many -strata, each -stratum homeomorphic (with respect to the usual Zariski topology) to the prime spectrum of a Laurent polynomial ring in finitely many commuting indeterminates, and each containing a unique -invariant prime ideal. Moreover, the primitive ideals of the algebra are precisely those that are maximal within their -stratum. For these reasons, an important first step towards understanding the prime and primitive spectra is to first study the -invariant prime ideals called -primes.

The deleting derivations algorithm of Cauchon [5, 6] has also proven quite useful. Roughly speaking, this procedure shows that when the -stratification theory applies to a given quantum algebra, one can often embed the set of -primes into the set of -primes of a quantum affine space. This is convenient since quantum affine spaces are typically easy to handle thanks to results of Goodearl and Letzter [12]. The strategy then is to reverse the deleting derivations procedure in order to transfer (more easily obtained) information about the quantum affine space back to information about the quantum algebra.

The -stratification and the deleting derivations theories both apply to quantum matrices in the generic case, i.e., when the parameter is a non-root of unity, and so a natural problem is to find generating sets for the -primes. For quantum matrices, this problem is fairly straightforward, yet even the case required a significant amount of work by Goodearl and Lenagan [10, 11]. However, in all cases their generating sets consisted of quantum minors and so it was conjectured that this held true in general. Launois [15, 16] was the first to prove this conjecture under the constraints and transcendental over . This was later extended to any of characteristic zero [8].

An important part of Cauchon’s results is a parametrization of the -primes of quantum matrices using what are now known in the quantum algebra community as Cauchon diagrams. It turns out that a Cauchon diagram encodes fundamental information about the corresponding -stratum. For example, the Krull dimension can be easily calculated from the Cauchon diagram using the main result of [1]. Launois also described an algorithm to find the generators of a given -prime from its Cauchon diagram, but the calculations involved very quickly become unwieldy. A graph theoretic interpretation of Launois’ algorithm provided in [4] forms the starting point for some of the results presented below. In fact, much of Section 3.1 may be seen as a combinatorial interpretation of the deleting derivations algorithm.

It is notable that Cauchon diagrams arose independently in work of Postnikov [19] in his investigations of the totally nonnegative Grassmannian. In this context, Cauchon diagrams are called L -diagrams (also Le-diagrams) and have been investigated by several authors (see Lam and Williams [14] and Talaska [21] in particular). The connections between these two areas and Poisson geometry have been explored by Goodearl, Launois and Lenagan [9, 8].

Finally, let us also mention that Yakimov [22, 23] has developed representation theoretic methods with great success. In particular, he has independently verified (and generalized) Goodearl and Lenagan’s conjecture, but again, only under the constraint that and transcendental over . Furthermore, the generating sets obtained are actually smaller than Launois’ in general. It is unclear how Yakimov’s work relates to the viewpoint presented in this paper, however, recent work of Geiger and Yakimov [7] explore the connections between Yakimov’s work and Cauchon’s, and so there is quite possibly a close relationship.

As will be reviewed in Section 2, the usual description of is by generators and relations. Our approach to is the focus of Section 3 where we begin by giving a directed graph and assign elements (“weights”) of a quantum torus to directed paths. We then discuss various subalgebras of the quantum torus generated by sums over path weights. In particular, Corollary 3.2.5 shows that quantum matrices can be so obtained. One nice aspect of this is that the quantum matrix relations naturally arise by considering intersecting paths (see the proofs of Theorem 3.1.12 and Theorem 3.2.3).

While at first it may appear that the description of quantum matrices “by paths” is a mere curiosity, it is in fact an indispensable tool in the bulk of this paper, Section 4. Here, the Goodearl-Lenagan conjecture is an immediate corollary to a stronger result, Theorem 4.4.1, which states that for any infinite field and non-root of unity , the quantum minors in a given -prime form a Gröbner basis with respect to a certain term ordering. The difficulty with this approach is that for a given -prime of , a priori we do not know any generating sets at all to which we can apply Buchberger’s algorithm, so we must check that the minors form a Gröbner basis by direct verification of the definition. The way we do this is by using the strategy noted above for the deleting derivations algorithm. That is, we transfer an (easily obtained) Gröbner basis for an -prime in a quantum affine space to a Gröbner basis for an -prime in quantum matrices.

Finally, many nonstandard terms and notation have been invented for use in this paper. An combined index and glossary is provided in an appendix to assist the reader in more easily locating the definitions should the need arise.

2. Quantum Matrices

Let us first set some data, notation and conventions that are to be used throughout this paper.

  • Fix: an infinite field , integers , and a nonzero, non-root of unity .

  • For a positive integer , we set

  • The set of matrices with integer entries is denoted by . The set of matrices with non-negative integer entries is denoted by .

  • The -entry of is denoted by , and is called the coordinate of this entry. In view of this, the elements of are called coordinates.

  • We often describe relative positions of coordinates using the usual meaning of terms such as north, northwest etc. For example, is northwest of if and , and north if and .

The restriction is made simply to avoid some inconveniences in various definitions that would occur if or . Fortunately, it is already known that all results presented in this paper hold when or since in these cases, all algebras in this paper reduce to quantum affine spaces, and such algebras can be dealt with using results of [12].

2.1. The Algebras

Definition 2.1.1.

The lexicographic order on is the total order obtained by setting

If , then denotes the largest element less than with respect to the lexicographic order.

Note 2.1.2.

Any reference in this paper relating to an ordering of the coordinates is with respect to the lexicographic order.

The algebras in the next definition each have a set of generators indexed by . It is natural to place these generators as the entries of an matrix that we call the matrix of generators.

Definition 2.1.3.

Let and set to be the smallest coordinate. Define to be the -algebra with the matrix of generators subject to the following relations. If

is any submatrix of , then:

  1. , ;

  2. , ;

  3. ;

Example 2.1.4.

If , and , then and has matrix of generators

The relations corresponding to Part 4 of Definition 2.1.3 are

The two extremities in the collection of are of the most interest to us.

Notation 2.1.5.

With respect to the notation in Definition 2.1.3:

  1. If , then in Part 4 of Definition 2.1.3 we always have

    We call this algebra quantum affine space, denoted . The entries of the matrix of generators of will often be labeled by for .

  2. If , then in Part 4 of Definition 2.1.3 we always have

    This algebra is the quantized coordinate ring of matrices over , denoted by and simply referred to as () quantum matrices.

  3. The localization of with respect to the multiplicative set generated by the standard generators is called the () quantum torus .

  4. Two elements will be said to -commute if there is an integer such that . Note that commuting elements -commute.

In later sections, we work intimately with monomials in the generators of , so we here set some notation in this respect. For the remainder of this section, fix and let be the matrix of generators for .

Notation 2.1.6.

If , then we write

written so that the indices obey the lexicographic order from smallest to largest as one goes from left to right. We call such a monomial a lexicographic term. Similar notation will be used both for the quantum torus (where ), and, if is the smallest coordinate, for (where all entries of are non-negative except possibly the -entry).

It is not difficult to check that each may be written as an iterated Ore extension which immediately yields the following.

Theorem 2.1.7.

The following properties hold for every .

  1. is a Noetherian domain.

  2. As a -vector space, has a basis consisting of the lexicographic terms with . The same properties also hold for the quantum torus (but with ).∎

Definition 2.1.8.

The lexicographic expression of is the unique linear combination of distinct lexicographic terms with . A lexicographic term in this expression will be called a lex term of .

For , we will require a slight extension of Theorem 2.1.7. Observe that any monomial in the standard generators of may be written as for some integer and lexicographic term . Since , the next result follows easily.

Proposition 2.1.9.

For any coordinate , the set of lexicographic monomials of involving only with is linearly independent over the subalgebra generated by the with . Moreover, for a set of monomials in the standard generators of , the following are equivalent.

  1. The set is linearly independent over .

  2. The set is linearly independent over .

  3. The matrices are distinct.

A similar set of statements hold for the quantum torus. ∎

We conclude this section by noting that has a natural -grading that will be very much exploited in the proof of Theorem 4.4.1. If

then the homogeneous component of degree is the subspace of spanned by the lexicographic monomials of the form , where satisfies

In other words, the sum of all entries in row of equals , and the sum of all entries in column of equals . All references in this paper to a grading on will be with respect to this grading.

2.2. The Deleting Derivations Algorithm

The relationship between and has been studied by Cauchon [6] as a special case of the more general theory developed in [5]. Here, we review his results as they apply to these algebras. For each result in this section, we fix with , let denote the smallest coordinate, and let be the matrix of generators of and the matrix of generators for

Theorem 2.2.1 (Cauchon [5], Lemme 2.1 and Théorème 3.2.1).
  1. The multiplicative set generated by is a left and right Ore set for , and the multiplicative set generated by is a left and right Ore set for .

  2. There is an injective homomorphism

    defined on the standard generators by

  3. There is an injective homomorphism

    defined on the standard generators by

  4. .∎

The homomorphism in Theorem 2.2.1 (2) is called the deleting derivations map. We call the homomorphism in Theorem 2.2.1 (3) the adding derivations map. (This map is called the “reverse deleting derivations map” in [15], and a step of the “restoration” algorithm in [9].)

The strategy of Cauchon’s theory is to use these maps to iteratively transfer information between and . For example, to embed the prime spectrum of the latter algebra into the prime spectrum of the former.

As usual, for an algebra , denote by the set of prime ideals, equipped with the Zariski topology. We may partition as



Theorem 2.2.2 (Cauchon [6], Section 3.1).

There exists an injective map

satisfying the following properties.

  1. Restricted to , is bijective, sending to

    If then

  2. Restricted to , is injective, sending to

    where is the unique homomorphism that maps the standard generators as

2.3. -Stratification

For many quantum algebras, including the , the structure of the prime spectrum may be understood by first understanding the prime ideals that are invariant under a rational action of an algebraic torus . For with matrix of generators , let and note that every induces an automorphism of by

Definition 2.3.1.

An -prime is a prime ideal such that for all . The set of all -primes of is denoted . The -stratum associated to an -prime is the set

Theorem 2.3.2 (Goodearl-Letzter [13] (or see [2], Part II)).

For every , there are finitely many -primes in , and

Remark 2.3.3.

Theorem 2.2.1 and Theorem 2.3.2 are where it is necessary to require to be a nonzero, non-root of unity. We also note here that the -primes are well-known to be homogeneous ideals.

The -primes of have generating sets of a simple form.

Theorem 2.3.4 (Goodearl-Letzter [12], Section 2.1(ii)).

A prime ideal is an -prime if and only if there exists a such that

It is convenient to describe these -primes by using diagrams.

Definition 2.3.5.

An diagram is an grid of squares, each square colored either black or white.

We index the squares of a diagram as one would the entries of an matrix. If

for some , then the diagram corresponding to as that in which the black squares are precisely those . Conversely, any diagram defines a subset corresponding to the indices of the black squares, and therefore a corresponding . We henceforth identify a diagram with the corresponding subset . Figure 1 presents two diagrams, the left one corresponding to the -prime .

Figure 1. Two diagrams.

The deleting derivations map behaves nicely with respect to -primes.

Theorem 2.3.6 (Cauchon [6], Section 3.1).

For every , , the map injects into . Consequently, the composition

is an injection of into .∎

In view of the strategy mentioned in Section 2.2, a natural problem is to identify the diagrams of those -primes in that are the image of an -prime in under . We call these Cauchon diagrams

Definition 2.3.7.

A diagram is a Cauchon diagram if, for any given black square, either every square to the left or every square above is also black.

The right diagram in Figure 1 is an example of a Cauchon diagram, while the left is not a Cauchon diagram since the black square in position has a white square both above and to its left.

Theorem 2.3.8 (Cauchon [6], Theéorème 3.2.2).

A diagram is a Cauchon diagram if and only if the corresponding -prime in is the image under of an -prime in

3. Quantum Matrices by paths

3.1. Graphs and Paths

Let be a Cauchon diagram and, by Theorem 2.3.8, consider the corresponding -prime of . With the notation of Section 2.3, the image of under the composition is an -prime of . The goal of this section is to explain how is isomorphic to a subalgebra of the quantum torus defined by considering paths in a directed graph that is defined using . In particular, when , we obtain a combinatorial description of .

Definition 3.1.1.

To a Cauchon diagram construct a directed graph called the Cauchon graph111“Cauchon graphs” already appear in [19] where they are called -graphs. We here call these Cauchon graphs to be consistent with the Cauchon diagrams from which they derive. as follows. The vertex set consists of white vertices

together with row vertices , and column vertices222There is ambiguity between labels of the row and column vertices, but the type of vertex we mean will always be explicitly stated. . The set of directed edges consists precisely of those in the following list.

  • If are distinct white vertices with and such that there is no white vertex for any , then we make an edge from to ;

  • If are distinct white vertices with such that there is no white vertex for any , then we make an edge from to ;

  • For we make an edge from to , where is the largest integer such that (if such a exists);

  • For we make an edge from to where is the largest integer such that (if such an exists).

Note 3.1.2.

There is a natural way to embed a Cauchon graph in the plane by placing it “on top” of the Cauchon diagram as follows. The white vertices are placed at the center of the corresponding white squares, the row vertices to the right of the corresponding diagram row, and the column vertices underneath the corresponding diagram column. An example is illustrated in Figure 2. We call this the standard embedding and always assume a given Cauchon graph is equipped with it. Hence, without confusion we can refer to aspects of a Cauchon graph using common directional or geometric terms333For example, horizontal, vertical, above, below, northwest, etc. That a diagram is a Cauchon diagram easily implies that the corresponding Cauchon graph has the following important property.







Figure 2. The graph , embedded on top of the Cauchon diagram .
Proposition 3.1.3.

The standard embedding of a Cauchon graph is planar. ∎

Definition 3.1.4.

A path in is a sequence of distinct vertices such that444strictly speaking, we are defining a directed path, but we will never have use for non-directed paths in this paper. for all , there exists an edge in directed from to . Naturally, we say that starts at and ends at and write .

We consider a directed edge from to to be a path and write . If is the edge between two consecutive vertices in a path , then we abuse notation by writing . Finally, if , , then we write to denote the concatenation of and . To a path in a Cauchon graph we will assign an element of the quantum torus as follows.

Definition 3.1.5.

Let be a Cauchon graph. Define the function

as follows, where the numbering and notation correspond to the edge types of Definition 3.1.1:

  • ;

  • ;

  • ;

  • .

The image of an edge is called the weight of .

If is a path, and , then the weight of is defined to be







Figure 3. The graph , with , and edges labeled by their weights. (Labels of white vertices omitted.)
Example 3.1.6.

Figure 3 illustrates the graph of Figure 2 with edges labeled by their weights. The weight of the path


It is convenient to observe that for a row vertex and a column vertex , the weight of a path can be computed by looking at the sequence of “turns”.

Definition 3.1.7.

Let be a path in a Cauchon graph starting from row vertex and ending at column vertex .

  • A -turn in is a white vertex such that the edge from to is horizontal, and the edge from to is vertical.

  • A L -turn in is a white vertex such that the edge from to is vertical and the edge from to is horizontal.

The next proposition follows easily using the definitions of edge and path weights.

Proposition 3.1.8.

Let be a path in a Cauchon graph where is a row vertex and is a column vertex. If is the subsequence consisting of all -turns and L -turns, then

Example 3.1.9.

For the path in Example 3.1.6, the vertex is a -turn, is a L -turn, and is a -turn, so that This, of course, agrees with Example 3.1.6.

Parts 1 and 2 of the next result are Lemmas 3.5 and 3.6 respectively in  [4]. Part 3 is proven similarly.

Lemma 3.1.10.

In a Cauchon graph , let be a white vertex, and row vertices with , and and column vertices with .

  1. If and are paths in with only in common, then

  2. If and are paths in with only in common, then

  3. If and are paths in with only in common, then

For the remainder of this section, fix and let be the smallest coordinate.

Notation 3.1.11.

For a row vertex and a column vertex of , let denote the set of all paths in for which no vertex larger than is a L -turn.

Figure 4. The shaded area represents all white vertices greater than the smallest coordinate . (This convention will be repeated in later illustrations.) In this example, , but .

Figure 4 is meant to clarify Notation 3.1.11, and while we have drawn a vertex in this figure, it will not exist if . The main theorem of this section is the following.

Theorem 3.1.12.

Let be a Cauchon graph, let be row vertices with , and let be column vertices.

  1. If , then there exists a permutation of sending where

  2. If , then there exists a permutation of sending where

  3. If , then there exists a permutation of sending where

  4. If , then:

    1. If , and , then

    2. There exists a bijective function from the subset of consisting of those with , to sending to where


Part 1: Let . Since , and have a last (white) vertex in common, say . See Figure 5. Therefore, we may write where and , and where and . Define and . We have and that and , i.e., the map is an involution and so a permutation.

Finally, we apply Lemma 3.1.10 to make our final conclusion as follows. If has only vertical edges, then

If has a horizontal edge, then

Figure 5. Illustration of Part 1 in the proof of Theorem 3.1.12. The left figure shows paths (solid) and (dashed). Right figure shows paths (solid) and (dashed).

Part 2: Let . In this case, and have a first common vertex, say . Therefore, we may write where and , and where and