]Peter Tingley and Ben Webster 0]0
Mirković-Vilonen polytopes and
|Peter Tingley||Ben Webster|
|Department of Mathematics and Statistics,||Department of Mathematics,|
|Loyola University Chicago||University of Virginia|
|ptingley @ luc.edu||bwebster @ virginia.edu|
Dedicated to the memory of Andrei Zelevinsky (1953-2013).
Abstract. We describe how Mirković-Vilonen polytopes arise naturally from the categorification of Lie algebras using Khovanov-Lauda-Rouquier algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of KLR algebras and MV polytopes.
MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense in finite type. Our construction on the other hand gives a map from the infinity crystal to polytopes for all symmetrizable Kac-Moody algebras. However, to make the map injective and have well-defined crystal operators on the image, we must in general decorate the polytopes with some extra information. We suggest that the resulting “KLR polytopes” are the general-type analogues of MV polytopes.
We give a combinatorial description of the resulting decorated polytopes in all affine cases, and show that this recovers the affine MV polytopes recently defined by Baumann, Kamnitzer and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.
- 1 Background
- 2 Cuspidal decompositions
- 3 KLR polytopes and MV polytopes
Let be a complex semi-simple Lie algebra. The crystal is a combinatorial object associated to the algebra . This crystal has an axiomatic definition, but many explicit realizations of it have appeared in the literature, and for many purposes it suffices to work with these. Here we consider the relationship between two such realizations:
the set is in canonical bijection with the set of simple gradable modules of Khovanov-Lauda-Rouquier (KLR) algebras, and
the set is in canonical bijection the set of Mirković-Vilonen polytopes.
This certainly defines a bijection between and , but does not describe it explicitly. One of our main results is a simple description of this bijection: There is a KLR algebra attached to each positive sum of simple roots. For any two such , there is a natural inclusion . Define the character polytope of an -module to be the convex hull of the weights such that .
The map is the unique crystal isomorphism from to .
We feel Theorem A is interesting in its own right, but perhaps more important is the fact that naturally indexes for any symmetrizable Kac-Moody algebra. Thus, one can try to use the map above to define Mirković-Vilonen polytopes outside of finite type. However, there are pairs of non-isomorphic simples with the same polytopes; for , in the notation of (3.6), this happens for and . Thus, the polytopes alone are not enough information to parametrize .
As suggested by Dunlap [Dun10] and developed in [BKT14], this problem can be overcome by decorating the edges of with extra information. In the current setting, the most natural data to associate to an edge is a “semi-cuspidal” representation of a smaller KLR algebra (see Definition 2.3). In complete generality, there are many different semi-cuspidal representations that can decorate a given edge, and we do not know a fully combinatorial description of the resulting object.
For edges parallel to real roots it turns out that there is only one possible semi-cuspidal representation, and so it is safe to leave off the decoration. Thus, in finite type the decoration is redundant.
Next consider the case when is affine of rank . Then the only non-real roots are multiples of , so the only edges of the polytope that must be decorated are those parallel to . The semi-cuspidal representations that can be associated to such an edge are naturally indexed by an -tuple of partitions (see Corollary 3.45). In fact, we can reduce the amount of information even further: as in [BKT14], the (possibly degenerate) -faces of parallel to are naturally indexed by the chamber coweights of an underlying finite type root system. Denote the face of corresponding to by . We in fact decorate with just the data of a partition for each chamber coweight (see Definition 3.46) in such a way that, for any edge parallel to
where are scalars attached to the facet defined in Definition 3.35. The representation attached to such an edge is determined in a natural way by .
Define an affine pseudo-Weyl polytope111In [BKT14], the analogous object is called a “decorated GGMS (Gelfand-Goresky-MacPherson-Serganova) polyotpe.” Since we work purely algebraicly without reference to the geometric structures studied in [GGMS87], we think it more appropriate to follow the usage of [Kam10, BK12], and use “pseudo-Weyl polytope.” to be a pair consisting of
a polytope in the root lattice of with all edges parallel to roots, and
a choice of partition for each chamber coweight of the underlying finite type root system which satisfies condition (0.1) for each edge parallel to .
To each representation of we associate its affine MV polytope (see Definition 3.48), which is a special decorated affine pseudo-Weyl polytope. Let be the set of these decorated polytopes. We seek a combinatorial characterization of . As in finite type, this can be done in terms of conditions on the 2-faces.
For every 2-face of an affine pseudo-Weyl polytope, the roots parallel to form a rank 2 sub-root system of either finite or affine type. If is of affine type, then generally has two edges parallel to , which are of the form and for unique chamber coweights . One naive guess is that we would obtain a rank-2 pseudo-Weyl polytope by decorating these imaginary edges with and , but this fails to satisfy (0.1), since and are too long. Instead, is the Minkowski sum of the line segment with a decorated pseudo-Weyl polytope , obtained by shortening and and decorating them with and . We will show that:
For an affine Lie algebra, the affine MV polytopes are precisely the decorated affine pseudo-Weyl polytopes where every 2-dimensional face satisfies
In [BKT14], analogues of MV polytopes were constructed in all symmetric affine types as decorated Harder-Narasimhan polytopes, and it was shown that these are characterized by their 2-faces. Thus Theorem B also allows us to understand the relationship between our decorated polytopes and those defined in [BKT14]:
Assume is of affine type with symmetric Cartan matrix. Fix and let be the corresponding element of . The affine MV polytope and the decorated Harder-Narasimhan polytope from [BKT14] have identical underlying polytopes. Furthermore, for each chamber coweight in the underlying finite type root system, the partition decorating as defined in [BKT14, Sections 1.5 and 7.6] is the transpose of our .
It is natural to ask for an intrinsic characterization of the polytopes in the general Kac-Moody case. We do not even have a conjecture for a true combinatorial characterization, since the polytopes are decorated with various semi-cuspidal representations, which at the moment are not well-understood. Some difficulties that come up outside of affine type are discussed in §3.7. However, our construction does still satisfy the most basic properties one would expect, as we now summarize (see Corollaries 3.10 and 3.11 for precise statements).
For an arbitrary symmetrizable Kac-Moody algebra, the map from to polytopes with edges labeled by semi-cuspidal representations is injective. Furthermore, for each convex order on roots, the elements of are parameterized by the possible tuples of semi-cuspidal representations of smaller KLR algebras decorating the edges along a corresponding path through the polytope, generalizing the parameterization of crystals in finite type by Lusztig data.
As we were completing this paper, some independent work on similar problems appeared: McNamara [McNa] proved a version of Theorem D in finite type (amongst other theorems on the structure of these representations) and Kleshchev [Kle14] gave a generalization of this to affine type. While there was some overlap with the present paper, these other works are focused on a single convex order, rather than giving a description of how different orders interact as we do in Theorems A, B and C.
We thank Arun Ram for first suggesting this connection to us, Joel Kamnitzer and Dinakar Muthiah for many interesting discussions, Monica Vazirani for pointing out the example of §3.6, Scott Carnahan for directing us to a useful reference on Borcherds algebras [Car], and Hugh Thomas for pointing out a minor error in our discussion of convex orders for infinite root systems. P. T. was supported by NSF grants DMS-0902649, DMS-1162385 and DMS-1265555; B. W. was supported by the NSF under Grant DMS-1151473 and the NSA under Grant H98230-10-1-0199.
Fix a symmetrizable Kac-Moody algebra . Let be its Dynkin diagram and its quantized universal enveloping algebra. Let be the Chevalley generators, and be the part of this algebra generated by the . Let be the weight lattice, the simple roots, the simple co-roots, and the pairing between weight space and coweight space.
We are interested in the crystal associated with . This is a combinatorial object arising from the theory of crystal bases for the corresponding quantum group This section contains a brief explanation of the results we need, roughly following [Kas95] and [HK02], to which we refer the reader for details. We start with a combinatorial notion of crystal that includes many examples which do not arise from representations, but which is easy to characterize.
(see [Kas95, Section 7.2]) A combinatorial crystal is a set along with functions (where is the weight lattice), and, for each , and , such that
increases by 1, decreases by 1 and increases by .
if and only if .
If , then .
We often denote a combinatorial crystal simply by , suppressing the other data.
A lowest weight combinatorial crystal is a combinatorial crystal which has a distinguished element (the lowest weight element) such that
The lowest weight element can be reached from any by applying a sequence of for various .
For all and all , .
Notice that, for a lowest weight combinatorial crystal, the functions and are determined by the and the weight of just the lowest weight element.
The following notion is not common in the literature, but will be very convenient.
A bicrystal is a set with two different crystal structures whose weight functions agree. We will always use the convention of placing a star superscript on all data for the second crystal structure, so , etc. We say that an element of a bicrystal is lowest weight if it is killed by both and for all .
We will consider one very important example of a bicrystal: along with the usual crystal operators and Kashiwara’s -crystal operators, which are the conjugates of the usual operators by Kashiwara’s involution (see [Kas93, 2.1.1]). The involution is a crystal limit of a corresponding involution of the algebra , but it also has a simple combinatorial definition in each of the models we consider.
The following is a rewording of [KS97, Proposition 3.2.3] designed to make the roles of the usual crystal operators and the -crystal operators more symmetric:
Fix a bicrystal . Assume and are both lowest weight combinatorial crystals with the same lowest weight element , where the other data is determined by setting . Assume further that, for all and all ,
For all ,
If then ,
If then and .
If then .
then , and , where is Kashiwara’s involution. Furthermore, these conditions are always satisfied by along with its operators .
We simply explain how [KS97, Proposition 3.2.3] implies our statement, referring the reader there for specialized notation. Define the map
One can check that our conditions imply all the conditions from [KS97, Proposition 3.2.3], so that result implies the crystal structure on defined by is isomorphic to . The remaining statements then follow from [KS97, Theorem 3.2.2]. ∎
The following is immediate from Proposition 1.4, but perhaps organizes the information in an easier way:
For any , and any , the subset of generated by the operators is of the form:
where the solid and dashed arrows show the action of , and the dotted or dashed arrows denote the action of . Here the width of the diagram at the top is , where is the bottom vertex (in the example above the width is 4). ∎
We will also make use of Saito’s crystal reflections from [Sai94].
Fix with . The Saito reflection of is . There is also a dual notion of Saito reflection defined by , or equivalently which is defined for those such that .
The operation does in fact reflect the weight of by , as the name suggests (although this fails if the condition does not hold).
Finally, we need the notion of string data for an element of . This appeared early on in the literature on crystals, implicitly in work of Kashiwara [Kas93] and more explicitly in work of Berenstein and Zelevinsky [BZ93]. It was also studied in in the context of KLR algebras (i.e. the context we use) in [KL09, §3.2] and [Web, §3.3].
Choose a list of simple roots in which each simple root occurs infinitely many times (for instance, one could choose an order on the roots and cycle).
For any the string data of with respect to is the lexicographically maximal list of integers such that .
Clearly all but finitely many of the must be zero in any given string datum. Note also that the element can easily be recovered from its string datum: .
1.2. Convex orders and charges
A convex order on roots is generally defined to be a total order such that, if and are all roots, then is between and . Here we need a more geometric definition, and we need to expand to have a notion of convex pre-orders. In fact, our definition makes sense for collections of vectors which do no necessarily come from root systems, and we will set it up in that generality. We will then see that in the case of finite type root systems our definition is equivalent to the usual one.
For this section, fix a finite dimensional vector space and a set of vectors in .
A convex preorder is a pre-order on such that,
For any equivalence class , any and any non-zero , we have that
For any equivalence class , any and any non-zero , we have that
A convex order is a convex pre-order which is a total order.
In the case of a total order, Definition 1.8 is equivalent to requiring that, for any such that for all ,
A pre-order on a countable set of vectors in a vector space is convex if and only if, for any equivalence class , there is a sequence of cooriented hyperplanes for such that for all , and each lies
on the positive side of for if and
on the negative side of for if .
We need to allow a sequence of hyperplanes because may be infinite.
Fix any finite subset of . Let denote the subsets consisting of vectors which are greater/less than according to . Consider the quotient , and the cones and in this space. Convexity implies that for all .
Any point in has a preimage in
which by convexity must in fact lie in . Hence .
Similarly, neither nor contains a line since if for , then and have preimages in which we can choose so that . Convexity thus implies that , so .
Thus, and are closed finite polyhedral cones in a finite dimensional vector space whose intersection consists exactly of the origin, neither of which contains a line. Two such cones are always separated by a hyperplane since their duals are full dimensional and span the whole space, and therefore contain elements in their interiors that sum to 0.
The preimage of this hyperplane in separates the elements of as desired; thus as we let grow, we will obtain the desired sequence of hyperplanes.
It is easy to see that, if such a sequence exists for every equivalence class , then the order must be convex. ∎
We now define charges, which are our main tool for constructing and studying convex orders.
A charge is a linear function such that the image is contained in some open half-plane defined by a line through the origin.
Every charge defines a preorder on by setting if and only if , where is the usual argument function on the complex numbers, taking a branch cut of which does not lie in the positive span of . This order is independent of the position of the branch cut. This preorder is clearly convex, and for generic , it is a total order.
Assume that is countable, that it does not contain any pair of parallel vectors, and that is contained in an open half-space . Then there is a convex total order on .
Choose a basis such that for lies in the hyperplane , and lies in .
We can define a charge by sending to elements of and to the upper half-plane. Since is countable, all the coefficients of in terms of lie in a countable subfield of . Choose and such that is linearly independent over , and consider the charge defined by . This sends no two elements of to points with the same argument, since otherwise, writing the two vectors as , we would have
for some . Comparing imaginary parts this is only possible if , so . This implies that for all by the linear independence of , so and are parallel, and we assumed does not contain parallel vectors. Thus, defines a total order. ∎
Assume that is countable, that it does not contain any pair of parallel vectors, and that lies in an open half-plane . Then every convex pre-order on can be refined to a convex order. Furthermore, this can be done by choosing any convex order on each equivalence class.
Fix a convex pre-order on and an equivalence class . By Lemma 1.13 we can choose a convex total order on . Let be the refinement of using the order on . Definition 1.8 clearly holds for any class which doesn’t lie in . Thus, we may reduce to the case where for some .
We need to show that if , then . We can write with and . If , then convexity implies that
On the other hand, if , then we have reduced to the same situation using only roots from , so it follows from the convexity of . ∎
Now fix a symmetrizable Kac-Moody algebra with root system and Cartan subalgebra . Let be the set of positive roots such that is not a root for any (this is all positive roots in finite type). From now on we will only consider convex orders on . In this case the conditions of Lemmata 1.13 and 1.14 clearly hold. Notice also that, since any root can be expressed as a non-negative linear combination of simple roots, for any convex total order the minimal and maximal elements must be simple.
We will need the following notion of “reflection” for convex orders and charges.
Fix a convex preorder such that is the unique lowest (reps. greatest) root. Define a new convex order by
and greatest (resp. lowest) for .
Similarly, for a charge such that is lowest (resp. greatest) amongst positive roots, define a new charge by .
It is straightforward to check that reflections for charges and convex orders are compatible in the sense that, for all charges such that is greatest or lowest, and coincide.
The following result is well known with the usual definition of convex order. The fact that it holds for our definition as well shows that the two definitions agree in the case of convex orders on finite type root systems.
Assume is of finite type. There is a bijection between convex orders on and expressions for the longest word , which is given by sending to the order
First, fix a reduced expression. It is well known that
is an enumeration of the positive roots, so we have defined a total ordering on positive roots. For any root , the hyperplane defined by the zeros of separates those larger than it from those smaller than it, so this order is convex by Lemma 1.10.
Now fix a convex order . The greatest root must be a simple root . The convex order as defined above also has a greatest root . Define in the same way using and continue as many times as there are positive roots. The list is a complete, irredundant list of positive roots. This implies that is a reduced expression for . Furthermore, if we apply the procedure in the statement to create an order on positive roots from this expression, we clearly end up with our original convex order. ∎
Of course, if is of infinite type, the technique in the proof of Proposition 1.16 will result not in a reduced word for the longest element (which does not exist), but an infinite reduced word in as well as a dual sequence constructed from looking at lowest elements. The corresponding lists of roots
are totally ordered, but don’t contain every root. We call the roots that appear in the list accessible from below, and those in the other list accessible from above. The terminology is due to the fact that the roots in the first list will correspond to edges near the bottom of the MV polytope, and those in the second list will correspond to edges near the top. The roots that are accessible from above or below are exactly those that are finitely far from one end of the order .
In the affine case, for most convex orders, only is neither accessible from above nor accessible from below; this happens exactly for the one-row orders from [Ito01], which includes all orders induced by charges. In more general types, one typically misses many roots, including many real roots. In many cases, one even misses simple roots.
Fix a convex order . For each and each real root which is accessible from above, define an integer by setting if is minimal for , and
for all other accessible from above roots.
Similarly, define for all which are accessible from below by if is maximal for , and
for other accessible from below roots.
We call the collection the crystal-theoretic Lusztig data for with respect to .
In infinite types, no root is accessible both from above and below, but in finite type all are. Thus, in order to justify our notation, we must prove that the two definitions we have given for agree. In fact, we now show that both agree with the exponents in Lusztig’s PBW basis element corresponding to for the reduced expression of giving the convex order . This connection explains the term “Lusztig data.”
In finite type, for any , the two definitions of in Definition 1.18 agree. Furthermore:
Let be the reduced expression for corresponding to the convex order . Then Lusztig’s PBW monomial corresponding to as in [Lus96, Proposition 8.2] is
For all , the geometric Lusztig data of the polytope corresponding to agrees with .
It follows by applying [Sai94, Proposition 3.4.7] repeatedly that the definitions of both read off the exponent of in Lusztig’s PBW monomial corresponding to for the order (with one definition one starts reading from the right of the monomial and with the other one starts reading from the left). Hence they agree and satisfy (i). It is shown in [Kam10, Kam07] that also satisfies (i), from which it is immediate that . ∎
The following notion of compatibility allows us to study an arbitrary convex order on using charges.
Fix a triple , where is a convex pre-order on , is an equivalence class for , and . A charge is said to be compatible if all roots in have the same argument with respect to and, for all of depth , we have if and only if and if and only if .
For every triple as in Definition 1.20 there is a compatible charge.
Choose a sequence of hyperplanes as in Lemma 1.10, and choose large enough that all ’s of depth are on the correct side of . One can choose a charge such that is the inverse image of the imaginary line, and such that some root has argument greater then . This must in fact be a compatible charge. ∎
Fix a convex pre-order and a cooriented hyperplane such that whenever is on the positive side of and is on the negative. Consider a simple root on the positive side of . Then there is a convex pre-order such that:
is the unique maximal root, and
for all with on the non-positive side of , if and only of and if and only if . That is, the relative position of any root on the non-positive side of with any other root remains unchanged.
If is a total order then can be taken to be a total order as well.
Let be any function whose vanishing locus is with the correct coorientation. Consider the function for . At , the only root on which has a positive value is ; for every other root on the positive side of , there is a unique such that .
and , or
and , or
This is a convex pre-order since all equivalence classes and initial/final segments are either defined as the vectors lying in or on one side of a hyperplane, or as equivalence classes or segments for . Certainly the relative position of any root on the negative side of with any other root agrees with the relative position for
By Lemma 1.14, if the order is total, we can refine to a total order by using to order within each equivalence class. ∎
1.3. Pseudo-Weyl polytopes
A pseudo-Weyl polytope is a convex polytope in with all edges parallel to roots.
For a pseudo-Weyl polytope , let be the vertex of such that is lowest, and the vertex where this is highest (these are vertices as for all roots ).
Fix a pseudo-Weyl polytope and a convex order on . There is a unique path through the 1-skeleton of from to which passes through at most one edge parallel to each root, and these appear in decreasing order according to as one travels from to .
Let be the minimal roots that are parallel to edges in , ordered by . Since is convex, for each , one can find such that for , and for . Let and . Construct a path in coweight space for ranging from to by, for for letting . As varies from to , the locus in the polytope where takes on its lowest value is generically a vertex of , but occasionally defines an edge. The set of edges that come up is the required path. ∎
Fix a pseudo-Weyl polytope and a convex order . For each , define to be the unique non-negative number such that the edge in parallel to is a translate of . We call the collection the geometric Lusztig data of with respect to .
Let be a pseudo-Weyl polytope and an edge of . Then there exists a charge such that is a total order and . In particular, a pseudo-Weyl polytope is uniquely determined by its geometric Lusztig data with respect to all convex orders coming from charges.
Since is an edge of , there is a functional such that
Since is a pseudo-Weyl polytope, is parallel to some root , and so . Furthermore, may be chosen so that for all other which are parallel to edges of . For any linear function such that define a charge by
For generic , the charge satisfies the required conditions. ∎
The following should be thought of as a general-type analogue of the fact that, in finite type, any reduced expression for can be obtained from any other by a finite number of braid moves. In fact, this statement can be generalized to include all convex orders, not just those coming from charges, but we only need the simpler version.
There is a natural height function on any pseudo-Weyl polytope given by . Thus for each face there is a notion of top and bottom vertices.
Let be a pseudo-Weyl polytope and two generic charges. Then there is a sequence of generic charges such that , and, for all , and differ by moving from the bottom vertex to the top vertex of a single 2-face of in the two possible directions.
Let be the set of root directions that appear as edges in . For , let . Clearly this is a charge. We can deform slightly, without changing the order of any of the roots in , such that
For all but finitely many , induces a total order on
For those where does not induce a total order, there is exactly one argument such that more then one root in has argument . Furthermore, the span of the roots with argument is 2 dimensional.
Denote the values of where does not induce a total order by . Fix with
Then is the required sequence. ∎
1.4. Finite type MV polytopes
Mirković-Vilonen (MV) polytopes are polytopes in the weight space of a complex-simple Lie algebra which first arose as moment map images of the MV cycles in the affine Grassmannian, as studied by Mirković and Vilonen [MV07]. Anderson [And03] and Kamnitzer [Kam10, Kam07] developed a realization of using these polytopes as the underlying set. Here we will not need details of these constructions, but will only use certain characterization theorems.
The following is discussed implicitly in [BK12].
For finite type , there is a unique map from to pseudo-Weyl polytopes such that
If is a convex order with minimal root , then, for all , , and .
If is a convex order with minimal root and , then, for all , and
Here is the Saito reflection from Definition 1.6. This map is the unique bicrystal isomorphism between and the set of MV polytopes.
The first step is to show that there is at most one map satisfying the conditions. To see this we proceed by induction. Consider the reverse-lexicographical order on collections of integers . Assume is minimal such that, for some convex order
and for two maps and satisfying the conditions, for all , but for some .
If we can reduce to a smaller such example using condition (ii). Otherwise, as long as some , we can reduce to a smaller such example using (iii). Clearly the map is unique if all , so this proves uniqueness.
We also need the following standard facts about MV polytopes:
Theorem 1.30 ([Kam10, Theorem D])
The MV polytopes are exactly those pseudo-Weyl polytopes such that all 2-faces are MV polytopes for the corresponding rank 2 root system. ∎
Theorem 1.31 ([Kam10, 4.2])
An MV polytope is uniquely determined by its geometric Lusztig data with respect to any one convex order on positive roots. ∎
1.5. Rank 2 affine MV polytopes
The and root systems correspond to the affine Dynkin diagrams
The corresponding symmetrized Cartan matrices are
Denote the simple roots by , where in the case of the short root is . Define for and for