Three combinatorial models for crystals, with applications to cylindric plane partitions
Abstract.
We define three combinatorial models for crystals, parametrized by partitions, configurations of beads on an “abacus”, and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible subcrystal corresponding to any dominant integral highest weight . Cylindric plane partitions actually parametrize a basis for , where is the space spanned by partitions. We use this to calculate the partition function for a system of random cylindric plane partitions. We also observe a form of rank level duality. Finally, we use an explicit bijection to relate our work to the Kyoto path model.
Contents
1. Introduction
This work was motivated by the Hayashi realization for crystals of level one representations, originally developed by Misra and Miwa [13] using work of Hayashi [5] (see also [2], Chapter 10). In that realization, the underlying set of the crystal consists of partitions, and the operators act by adding a box to the associated Young diagram. We wondered if there was a similar realization for representations of arbitrary level , where the operators would act by adding an ribbon (see Section 2.1). It turns out that there is. To prove that our construction works, we need a second model, which is based on the abacus used by James and Kerber ([7], Chapter 2.7). The abacus model is reminiscent of a “Dirac sea,” and we think it is interesting in its own right.
The crystals one obtains from the abacus model are not irreducible. However, one can pick out a “highest” irreducible subcrystal, so we do have a model for the crystal of any irreducible integrable highest weight representation of . There is also a reducible subcrystal whose underlying set is naturally in bijection with the set of cylindric plane partitions with a given boundary. This is our third combinatorial model. We also define a action on the space spanned by cylindric plane partitions, which commutes with our crystal operators. Furthermore, if we use both actions, the space becomes irreducible; in fact, the set of cylindric plane partitions with a given boundary forms a basis for , where is an irreducible representation of some , and is the space spanned by all partitions.
In this picture, the weight of a cylindric plane partition is easily calculated from the principally graded weight of the corresponding element of and the size of the corresponding partition . This allows us to calculate the partition function for the system of random cylindric plane partitions studied by Borodin in [3]. Our answer looks quite different from the formula given by Borodin, but we can directly show that they agree. This gives a new link between Borodin’s work and the representation theory of .
There is a symmetry in our model which allows us to consider a given cylindric plane partition as an element of either or , where is some level highest weight for , and is a level highest weight for determined by . This implies an identity of characters, which is our ranklevel duality. Our result is similar to a duality discovered by Frenkel [4]. We show how these are related and obtain a new proof of Frenkel’s result.
We then relate our work to the Kyoto path model developed by Kashiwara et. al. in [10] and [11] (see [6] for a more recent explanation). We do this by exhibiting an explicit crystal isomorphism between the highest irreducible component of the abacus model and the Kyoto path model for a particular perfect crystal and ground state path.
We finish with some questions. Most notably, it is natural to ask if our crystal structures can be lifted to get representations of . We believe that there should be such a lifting for the space spanned by cylindric plane partitions. This should be similar to the deformed Fock space studied by Kashiwara, Miwa, Petersen and Yung in [12].
Before beginning, we would like to mention a 1991 paper by Jimbo, Misra, Miwa and Okado [8] which contains some results relevant to the present work. In particular, they present a similar realization for the crystal of any irreducible integrable highest weight representation of .
1.1. Acknowledgments
I would like to thank Mark Haiman, Tony Chiang, Brian Rothbock, Alex Woo, Sami Assaf, and everyone else who attended Mark’s seminar in 20032004. This paper could never have happened without all of your input. I would also like to thank Alexander Braverman, Alejandra Premat, Anne Schilling and Monica Vazirani for useful discussions. Finally, I would like to thank my advisor Nicolai Reshetikhin for his patience and support.
1.2. Change log
The purpose of this section is to record significant changes to this work since publication (v2).
 Aug 26, 2008:

Section 4.2 was corrected. With Definition 4.9 as stated in the published paper, the crystal structure on cylindric plane partitions described in Section 4.2 and the caption to Figure 13 was incorrect. Note that the crystal structure on descending abacus configurations described in Section 3 was correct. However we incorrectly translated this into the language of cylindric plane partitions. Section 4.2 is independent of the rest of the paper, so the rest of our results remain true as originally stated.
2. Background
In this section we review some tools we will need. We only include those results most important for the present work, and refer the reader to other sources for more details.
2.1. The abacus
Here we explain the abacus used by James and Kerber in [7]. We start by defining a bijection between partitions and rows of beads. This is essentially the correspondence between partitions and semiinfinite wedge products (see for example [9], Chapter 14); the positions of the beads correspond to the factors in the wedge product. As in [7], this one row is transformed into several parallel rows of beads on an abacus.
We use the “Russian” diagram of a partition, shown in Figure 1. Place a bead on the horizontal axis under each downsloping segment on the edge of the diagram. The corresponding row of beads uniquely defines the partition. Label the horizontal axis so the corners of all boxes are integers, with the vertex at . For a partition , where we define for large , the positions of the beads will be . The empty partition corresponds to having beads at all negative positions in , and none of the positive positions. Adding a box to the partition corresponds to moving a bead one step to the right.
Recall that a ribbon is a skew partition whose diagram is connected and has at most one box above each position on the horizontal axis (i.e. it is a strip one box wide). Adding an ribbon (that is, a ribbon with boxes) to a partition moves one bead exactly steps to the right in the corresponding row of beads, possibly jumping over other beads. See Figure 2.
It is often convenient to work with just the “bead” picture. In order to avoid confusion, we denote the empty spaces by white beads, and indicate the position of the origin by a line. The example shown in Figure 1 becomes:
We then put the beads into groups of starting at the origin, as shown below for :
Rotating each group 90 degrees counterclockwise, and compressing, we get rows:
We will call this the level abacus (in this example ). Adding an ribbon now corresponds to moving one bead forward one position, staying on the same row. For instance, adding the ribbon as in Figure 2 corresponds to moving the third black bead from the right on the top row, which gives:
As explained in ([7] Chapter 2.7), this model immediately gives some interesting information about the partition: The rows can be interpreted as partitions, using the correspondence between partitions and rows of beads (shifting the origin if necessary). This is known as the quotient of . We can also consider the partition obtained by pushing the beads on each row as far to the left as they will go, but not changing rows (and only doing finitely many moves). This is the core.
2.2. Crystals and tensor products
We use notation as in [6], and refer the reader to that book for a detailed explanation of crystals. For us, a crystal is a set associated to a representation of a symmetrizable KacMoody algebra , along with operators and , which satisfy some conditions. The set records certain combinatorial data associated to , and the operators and correspond to the Chevalley generators and of . If is the crystal of an irreducible, integral highest weight module , then corresponds to a canonical basis for . That is, we can associate to each a , such that is a basis for .
Often will be represented as a colored directed graph whose vertices are the elements of , and we have a colored edge from to if . This records all the information about , since if and only if .
The tensor product rule for modules leads to a tensor product rule for crystals, which we will now review. We then present an equivalent definition of the tensor product rule using strings of brackets. This fits more closely with our later definitions, and also helps explain why many realizations of crystals (for example, the realization of crystals using Young tableaux) make use of brackets.
We start by defining three elements in the root lattice of associated to each element . Let be a symmetrizable KacMoody algebra, with simple roots indexed by . Let the crystal of an integrable representation of . For each and , define:
These are always finite because is integrable.
Definition 2.1.
Let be the fundamental weight associated to . For each , define three elements in the weight lattice of by:



.
Comment 2.2.
It turns out that will always be equal to the weight of the corresponding canonical basis element (see for example [6]).
We now give the tensor product rule for crystals, using conventions from [6]. If and are two crystals, the tensor product is the crystal whose underlying set is , with operators and defined by:
(1) 
(2) 
This can be reworded as follows. In this form it is known as the signature rule:
Lemma 2.3.
For , let be the string of brackets , where the number of is and the number of is . Then the actions of and on can be calculated as follows:
Proof.
This formula for calculating and follows immediately from Equations (1) and (2). To see that if there are any uncanceled , notice that in this case you always act on a factor that contributes at least one , and hence has . By the definition of , does not send this element to . The proof for is similar. ∎
The advantage of Lemma 2.3 over equations (1) and (2) is that we can easily understand the actions of and on the tensor product of several crystals:
Corollary 2.4.
Let be crystals of integrable representations of . Let . For each , let be the string of brackets , where the number of is and the number of is . Then:



is the number of uncanceled in .

is the number of uncanceled in .
Proof.
Parts (i) and (ii) follow by iterating Lemma 2.3. To see part (iii), notice that, if the first uncanceled is in , then and acts on . Hence changes by reducing the number of by one, and increasing the number of by one. The only affect on is that the first uncanceled is changed to . This reduces the number of uncanceled by one. will send the element to exactly when there are no uncanceled left. Hence part (iii) follows by the definition of . Part (iv) is similar. ∎
3. Crystal structures
In this section, we define two families of crystal structures for . In the first, the vertices are partitions. In the second, the vertices are configurations of beads on an abacus. In each case the family is indexed by a positive integer . We will refer to as the level of the crystal, since, using the results of this section, it is straightforward to see our level crystal structure decomposes into a union of crystals corresponding to level irreducible representations of . We will see the crystal graph of every irreducible representation appear as an easily identifiable subcrystal of the abacus model. We do not see every irreducible representation using the partition model as we define it, although we can get the others using a simple change of conventions (just shift the coloring in Figure 3). There is a slight subtlety in the case of , since the proof of Theorem 3.1 fails. Theorem 3.14 and the results from Section 4 do hold in this case, as we show in Section 5 using the Kyoto path model. However, we cannot prove the full strength of Theorem 3.1 for .
3.1. Crystal structure on partitions
We now define level crystal operators for , acting on the set of partitions (at least when ). Figure 3 illustrates the definition for and : Color the boxes of a partition with colors , where all boxes above position on the horizontal axis are colored for modulo . To act by , place a above the horizontal position if boxes in that position are colored , and you can add an ribbon whose rightmost box is above . Similarly, we put a above each position where boxes are colored and you can remove an ribbon whose rightmost box is above . acts by adding an ribbon whose rightmost box is below the first uncanceled from the left, if possible, and sending that partition to if there is no uncanceled . Similarly, acts by removing an ribbon whose rightmost box is below the first uncanceled from the right, if possible, and sending the partition to otherwise.
Theorem 3.1 below will imply that, for , these operators do in fact endow the set of partitions with an crystal structure.
3.2. Crystal structure on abacus configurations
Since we can identify partitions with certain configurations of beads on the abacus (abacus configurations), the operators and defined above give operators on the set of abacus configurations coming from partitions. We know from Section 2.1 that adding (removing) an ribbon to a partition corresponds to moving one bead forward (backwards) one position in the corresponding strand abacus. This allows us to translate the operators and to the abacus model. We actually get a definition of operators and on the set of all abacus configurations, regardless of whether or not they actually come from partitions (see Figure 4):
Color the gaps between the columns of beads with colors, putting at the origin, and in the gap, counting left to right. We will include the colors in the diagrams by writing in the ’s below the corresponding gap. The operators and are then calculated as follows: Put a every time a bead could move to the right across color , and a every time a bead could move to the left across . The brackets are ordered moving up each colored gap in turn from left to right. We group all the brackets corresponding to the same gap above that gap. moves the bead corresponding to the first uncanceled “(” from the left one place forward, if possible, and sends that element to 0 otherwise. Similarly, moves the bead corresponding the the first uncanceled “)” from the right one space backwards, if possible, and sends the element to otherwise.
We will now show that the operators and give the set of abacus configurations the structure of an crystal, when . It follows immediately that our level operators on the set of partitions also gives us a crystal structure, since the map sending a partition to the corresponding abacus configuration preserves the operators and .
Theorem 3.1.
Fix and . Define a colored directed graph as follows: The vertices of are all configurations of beads on an strand abacus, which have finitely many empty positions to the left of the origin, and finitely many full positions to the right of the origin. There is a colored edge from to if and only if . Then each connected component of is the crystal graph of some integrable highest weight representation of .
Proof.
By [10], Proposition 2.4.4 (see also [14]), it is sufficient to show that, for each pair , each connected component of the graph obtained by only considering edges of color and is
Choose some abacus configuration , and some , and consider the connected component containing of the subgraph of obtained by only considering edges colored and . If , then and clearly commute (as do and ). Also, it is clear that if we only consider one color , then is a disjoint union on finite directed lines. This is sufficient to show that the component containing is an crystal, as required.
Now consider the case when . The abacus model is clearly symmetric under shifting the colors, so, without loss of generality, we may assume and . Also, we need only consider the abacus for , since if there are columns of beads not bordering a or gap, they can be ignored without affecting or . As shown in Figure 5, each connected component of this crystal is the crystal of an integrable representation, as required. ∎
3.3. More on the abacus model, including the highest irreducible part
This section is a little technical. We introduce some definitions, then present some results about the structure of the crystals defined in Section 3.2. Lemma 3.12 is the main result we need for the applications in Section 4. We state and prove this lemma without using the fact that this is an crystal, so it holds even for , when the proof of Theorem 3.1 fails. We finish the section by identifying the highest irreducible part, which gives us a realization of the crystal graph for any irreducible integrable representation of of level . We suggest the casual reader look at the statements of Lemma 3.12 and Theorem 3.14, and refer to the definitions as needed; the details of the proofs can safely be skipped.
Label the rows of the abacus , where is at the bottom, 1 is the next row up, and so on. For an abacus configuration , we denote the row of by , and the position of the bead on that row, counting from the right, by . We will always assume that any row we are considering has only finitely many empty negative positions, and finitely many full positive positions. That is, it can only differ in finitely many places from the row corresponding to the empty partition.
Definition 3.2.
Definition 3.3.
Let be a row of beads with finitely many negative positions empty and finitely many positive positions full. For each , let denote the positions of the bead in , counting from the right.
Definition 3.4.
Let and be two rows of beads. We say if for all .
Definition 3.5.
Let be an abacus configuration. Define a row of beads for any , by letting be the row of the abacus if (counting up and starting with 0), and extending to the rest of using . That is, is , but shifted steps to the left.
Definition 3.6.
We say a configuration of beads is descending if . Equivalently, is descending it if for all .
Comment 3.7.
Notice that a descending abacus configuration satisfies for all . Thus, if we only draw rows of the abacus, each is at most columbs wide (see Figure 6).
Definition 3.8.
The tightening operator is the operator on abacus configuration which shifts the bead on each row down one row, if possible. Explicitly:
See Figure 6 for an example. Similarly shifts the bead on each row up one row:
Definition 3.9.
We use the notation to denote the set consisting of the black bead from the right on each row.
Definition 3.10.
A descending configuration of beads is called tight if, for each , the beads are positioned as far to the left as possible, subject to the set being held fixed, and, for each , . See Figure 6. This is equivalent to saying for all , since we only deal with descending configurations, so is the only way to move closer to .
Lemma 3.11.
The set of descending abacus configurations (along with 0) is closed under the operators and defined in Section 3.2. Furthermore, the restriction of and to the set of descending abacus configurations can be calculated using the following rule:
For each , let be the string of brackets where the number of is the number of beads in in position , and the number of is the number of beads in in position . Let .
If the first uncanceled from the right in comes from , then moves a bead in one step to the left. The bead that moves is always the last bead of in position that you encounter moving up the columns in turn from left to right. If there is no uncanceled in , then sends that element to zero.
If the first uncanceled from the left in comes from , then moves a bead in one step to the right. The bead that moves is always the first bead of in position that you encounter moving up the columns in tun from left to right. If there is no uncanceled in , then sends that element to zero.
Proof.
Let be the string of brackets used to calculate and in Section 3.2. The descending condition implies:

All the brackets in coming from beads in always come before all the brackets coming from beads in .

All the in coming from always come before all the in coming from
These facts are both clear if you first do moves as in Figure 7 until the first bead of in position is on the bottom row of the abacus. As argued in the caption, these moves commute with the actions of and , and preserves the set of descending abacus configurations.
Let be the substring of consisting of brackets coming from . Then is obtained from by simply adding a string of canceling brackets between each and , where the number of and is the number of pairs of touching beads, one in and one in , that are in positions and respectively. Inserting canceling brackets does not change the first uncanceled . Hence the first uncanceled in will come from a bead in if and only if the first uncanceled in comes from a bead in . Therefore our new calculation of moves a bead in the right . It remains to show that the calculation of using always moves the first bead of is position . But this follows immediately from property (ii) above. The proof for is similar. Hence the new rule agrees with our definition of and .
This new rule clearly preserves the set of descending abacus configurations. ∎
We are now ready to state and prove our main lemma concerning the structure of the operators and acting on abacus configurations:
Lemma 3.12.
Fix , and . Consider the set of strand abacus configurations, colored with as shown in Figure 4. Let be the colored directed graph whose vertices are all descending abacus configurations with a given compactification , and where there is an edge of color from to if and only if . Then:

The operators and , restricted to the set of descending abacus configurations, commute with and .

The sources of (i.e. vertices that are not the end of any edge) are exactly those configurations that can be obtained from by a series of moves for various .

The set of tight descending abacus configurations is a connected component of .

If we add an edge to connecting and whenever for some , then is connected.
Proof.
(i): Calculate as in Lemma 3.11. Then clearly does not change the string of brackets , and hence commute with , as long as and . The only potential problem is if but (or visa versa).
Assume but . Then must move the only bead of that hits when you apply . must be the first bead of in position (moving up the columns and left to right). The last bead of in position must be on the the row below . Since hits when you apply , this must also be the last row of containing a bead in position . By Lemma 3.11, for to move, there must be at least as many beads of in position as beads in in position . The descending condition then implies that the number of beads of in position must be equal to the number of beads of in position . This is illustrated below:
But then is in fact zero, since even after applying the bottom bead of in position cannot be shifted down without hitting So the problem cannot in fact occur. The other case is similar, as are the cases involving or .
(ii): It is clear from part (i) that applying operators to will always give a source (as long as the configuration is not sent to 0). So, let be a source, and we will show that can be tightened to by applying a series of operators for various . First, for each , define:
(3)  
(4) 
We will use the notation (respectively ) to mean the coefficient of in (respectively ). For each , define to be the abacus configuration obtained by removing the first black beads on each row, counting from the right. When we calculate as in Lemma 3.11, all the brackets from always come before the brackets from , for any . Hence, if is a source, then so is for all . We will prove the following statement for each :
(5) 
can differ in only finitely many places from a compact configuration, so is compact for large enough , and (5) clearly holds for compact configurations. We proceed by induction, assuming (5) holds for some and proving it still holds for . Since the rule for calculating implies that . Since is still a source, we must have for each . But . Hence we see that . So, (5) holds for .
By Definition 3.10 and the definitions of and , (5) implies that we can tighten each until it is right next to . Therefore, can be tightened to a compact configuration, which must be by the definition of . Part (ii) follows since if and only if .
(iii): The graph is graded by , where the degree of is the number of times you need to move one bead one step to the left to reach . Every connected component to has at least one vertex of minimal degree. must be a source, since each is clearly degree 1. Hence each connected component of contains a source. By part (ii), we can tighten any source to get . Hence the set of tight descending abacus configurations in contains exactly one source, namely . By Lemma 3.11 and part (i), the set of tight descending abacus configurations is closed under the operators , so it is a complete connected component of .
Definition 3.13.
Let be a compact, descending abacus configuration. Define a dominant integral weight of by
where is the number of such that the last black bead of is in position modulo n. Equivalently,
Note that uniquely determines , up to a transformation of the form for some (see Definition 3.5). That is, up to a series of moves as in Figure 7.
Theorem 3.14.
Fix and , and let be a compact, descending configuration of beads on an strand abacus colored with as shown in Figure 4. Let be the colored directed graph whose vertices are all tight, descending abacus configurations with compactification , and there is a colored edge from to if . Then is a realization of the crystal graph for the representation .