Crystal energy functions via the charge in types A and C
CCRYSTAL ENERGY FUNCTIONS VIA THE CHARGEIN TYPES A AND C CRISTIAN LENART AND ANNE SCHILLING
Abstract.
The Ram–Yip formula for Macdonald polynomials (at t = 0) provides a statisticwhich we call charge. In types A and C it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we prove that the charge is equal to the(negative of the) energy function on affine crystals. The algorithm for computing charge ismuch simpler and can be more efficiently computed than the recursive definition of energy interms of the combinatorial R -matrix. Introduction
The energy function of affine crystals is an important grading used in one-dimensional con-figuration sums [10, 11] and generalized Kostka polynomials [43, 45, 46]. It is defined by theaction of the affine Kashiwara crystal operators through a local combinatorial rule and the R -matrix.From a computational perspective, the definition of the energy is not very efficient, as itinvolves a recursive definition of a local energy, and also the combinatorial R -matrix, for whichnot in all cases efficient algorithms exist. This leads us to the role of the charge statistic,which can be calculated very efficiently, as it only involves the detection of descents and thecomputation of arm lengths of cells in Young diagrams.Charge was originally defined in type A by Lascoux and Sch¨utzenberger [20] as a statisticon words with partition content. It is calculated by counting certain cycles in the given word;see Section 3. Lascoux and Sch¨utzenberger showed that the charge can also be defined asthe grading in the cyclage graph, and used it to express combinatorially the Kostka–Foulkespolynomials, or Lusztig’s q -analogue of weight multiplicities [34], based on their Morris recur-rence. In type A , Nakayashiki and Yamada [37] analyzed the subtle combinatorial relationshipbetween charge and the R -matrix, showing that the energy coincides with the charge. In [19]it was observed that the cyclage is related to the action of the crystal operator f on a tensorproduct of type A columns (by the Kyoto path model, the latter can be identified with an affineDemazure crystal). Thus, the results of Lascoux–Sch¨utzenberger and Nakayashiki–Yamada arerederived in a more conceptual way. See also the work of Shimozono [45, 46] for a more extensivediscussion of the combinatorics involved in [19], in the more general context of tensor productsof type A Kirillov–Reshetikhin (KR) crystals of arbitrary rectangular shapes, as opposed toonly column shapes. Charge for KR crystals of rectangular shape (or Littlewood–Richardsontableaux) was also defined in [43] using cyclage.
Mathematics Subject Classification.
Primary 05E05. Secondary 33D52, 20G42.Partially supported by the Research in Pairs program of Mathematiches Forschungsinstitut Oberwolfach, theNSF grant DMS–1101264, and an Individual Development Award from SUNY Albany.Partially supported by the NSF grants DMS–0652641, DMS–0652652, DMS–1001256, the “Research in Pairs”program by the Mathematiches Forschungsinstitut Oberwolfach in 2011, and the Hausdorff Institut in Bonn. a r X i v : . [ m a t h . C O ] J a n C. LENART AND A. SCHILLING
Lecouvey [24, 25] extended two approaches to the Lascoux–Sch¨utzenberger charge, namelycyclage and catabolism, to types B , C , and D . He thus defined a charge statistic on thecorresponding Kashiwara–Nakashima (KN) tableaux [17]. But he was only able to relate hischarge to the corresponding Kostka–Foulkes polynomials in very special cases, as the originalidea of Lascoux–Sch¨utzenberger based on the Morris recurrence, which he pursued, has limitedapplicability in this case.In this paper we use a charge statistic coming from the Ram–Yip formula [40] for Macdonaldpolynomials P µ ( x ; q, t ) of arbitrary type [35] at t = 0. The terms in this formula correspond tocertain chains of Weyl group elements which come from the alcove walk model (this was definedin [3, 32, 33], and was then developed in subsequent papers, including [40]). The statistic isdefined on the mentioned chains, and describes the powers of q . In [30] it is shown that, intypes A and C , the chains are in bijection with elements in a tensor product of KR crystals ofthe form B k, . It is also shown that, under this bijection, the above statistic can be translatedinto a statistic on the elements of the mentioned crystal, which we call charge. Thus, we have(1.1) P µ ( x ; q,
0) = (cid:88) b ∈ B µ (cid:48) , ⊗ B µ (cid:48) , ⊗ ... q charge( b ) x wt( b ) . In type A , one can rewrite this formula as an expansion of the Macdonald P -polynomials interms of Schur polynomials s λ ( x )(1.2) P µ ( x ; q,
0) = (cid:88) λ K λ (cid:48) µ (cid:48) ( q ) s λ ( x ) ;here K λ (cid:48) µ (cid:48) ( q ) is the Kostka–Foulkes polynomial and λ (cid:48) denotes the transpose of the partition λ .A generalization of (1.2) to simply-laced types was given in [12]; in types A and D , this result issharpened in [42, Section 9.2] by replacing the Kostka–Foulkes polynomials with the correspond-ing one-dimensional configuration sums (which are generating functions for the energy). Both(1.1) in type A and (1.2) are expressed combinatorially in terms of the Lascoux–Sch¨utzenbergercharge, whereas the type C charge given by (1.1) is a new statistic. It is worth noting thatthe main ingredient in these charge constructions is the quantum Bruhat graph [2], which firstarose in connection to Chevalley multiplication formulas for the quantum cohomology of flagvarieties.Due to Schur–Weyl duality, weight multiplicities and tensor product multiplicities are bothgiven by the Kostka matrix in type A . In fact, the Kostka–Foulkes polynomials are equal toLusztig’s q -analogue K of weight multiplicities as well as the one-dimensional configurationssums X , which are q -analogues of tensor product multiplicities. For other types, such corre-spondences are only known for large rank (or in the stable case); see for example [26, 28, 50].As is evident from Equation (1.1) the charge we are concerned with in this paper is naturallyrelated to q -analogues of tensor product multiplicities or the one-dimensional sums X .The goal of this paper is to show in an efficient, conceptual way that the charge in [30]coincides with the energy function on the corresponding tensor products of KR crystals. Wefocus on types A and C , and expect to extend these results to types B and D . There areseveral reasons for not yet addressing the case of arbitrary types. First of all, the elementsof the crystals B k, of classical types are represented in a very explicit way, by KN columns.In fact, concrete descriptions of certain KR crystals of exceptional type have only been givenin special cases; see for example [13, 16, 36, 51]. Furthermore, the stable one-dimensionalconfiguration sums studied in [28] are all of classical type. However, we do expect that themain result of this paper would generalize to arbitrary types, if we were to phrase it in the RYSTAL ENERGY VIA CHARGE 3 context of the alcove walk model and the statistic in the Ram–Yip formula mentioned above(see Section 7.3).We use the recent reinterpretation in [42] of the (global) energy function as the affine gradingon a tensor product of KR crystals under Demazure arrows (see Definition 2.7). In type A , KRcrystals are perfect and hence, by the Kyoto path model [14, 15], can be realized as Demazurecrystals. By the result of [42], Demazure arrows change energy by 1. Together with the resultthat charge is well-behaved under crystal operators, this proves the equality of energy andcharge. For type C , we use the same approach, but in this case KR crystals are not perfect.There is still an embedding of a tensor product of KR crystals into an affine highest weightcrystal (see Proposition 2.9) by analogy with the Kyoto path model, but now there are severalhighest weight components in the image, instead of just one. For each of these components weexhibit an explicit path from its highest weight (or ground state) to “type A elements” in thecomponent, using only Demazure arrows (see Theorem 5.2). This additional result suffices toestablish the equality of energy and charge in type C , based on the corresponding result in type A . As a by-product, we obtain an explicit description of the components that appear in thenonperfect setting of single columns for type C .Our main result can now be stated as follows. Theorem 1.1.
Let B = B r N , ⊗ · · · ⊗ B r , be a tensor product of KR crystals in type A (1) n − ortype C (1) n with r N ≥ r N − ≥ · · · ≥ r ≥ . Then for all b ∈ B we have (1.3) D ( b ) = − charge( b ) , where D ( b ) is given in Definition , and charge( b ) is defined in Section . From a theoretical point of view, the above result is not surprising due to work of Ion [12],which relates Macdonald polynomials at t = 0 and affine Demazure characters in simply-lacedtypes. However, this result does not work in type C ; in addition, it only gives an equalityof polynomials (the generating functions for the statistics and the weights), not of individualterms.To compare our work with the previous papers on charge and the energy, let us first saythat our results apply to arbitrary vertices in a tensor product of KN columns, not just tothe highest weight elements (with respect to the nonzero arrows), that are used in the workinvolving Kostka–Foulkes polynomials. In type A , our approach via affine Demazure crystalscomes closest to [19, 45, 46]. However, in addition to the aspect mentioned above, it differsfrom the approach in these papers because we do not use the cyclage operation, which is basedon the corresponding plactic relations; see [24]. These relations are the main cause of thecomplications in type C , in the work of Lecouvey [24, 25].The paper is organized as follows. In Section 2 we review the necessary crystal theory anddefine the energy function. In Section 3 we give the definition of charge both in types A and C . The proof of Theorem 1.1 for type A using the method of Demazure arrows is given inSection 4. In Section 5 we classify the various components under the Demazure arrows in type C in the extension of the Kyoto path model, and show that each ground state is connectedto a type A filling. These results are used in Section 6 to prove Theorem 1.1 for type C . Weconclude in Section 7 with a discussion of various possible extensions of this work. Acknowledgements.
We would like to thank the Mathematisches Forschungsinstitut in Ober-wolfach (MFO), Germany, for their support. The breakthrough in this project occurred during
C. LENART AND A. SCHILLING
Research in Pairs at MFO in February 2011. Many of our results were found exploring withcode on crystals in
Sage [47, 48].We would also like to thank S. Gaussent, M. Kashiwara, C. Lecouvey, P. Littelmann,S. Morier-Genoud, M. Okado, and P. Tingley for helpful discussions.2.
Crystals and energy function
In this section we review and set up crystal theory and define the energy function.2.1.
Crystal generalities.
Crystal bases provide a combinatorial method to study represen-tations of quantum algebras U q ( g ). For a good review on crystal base theory see the book byHong and Kang [9]. Here g is a Lie algebra or affine Kac–Moody Lie algebra with index set I ,weight lattice P , and simple roots α i with i ∈ I . The set of dominant weights is denoted by P + . For affine Kac–Moody (resp. finite Lie) algebras we denoted the fundamental weights byΛ i (resp. ω i ) for i ∈ I .A g - crystal is a nonempty set B together with maps e i , f i : B → B ∪ {∅} for i ∈ I andwt : B → P . For b ∈ B , we set ε i ( b ) = max { k | e ki ( b ) (cid:54) = ∅} , ϕ i ( b ) = max { k | f ki ( b ) (cid:54) = ∅} , ε ( b ) = (cid:88) i ∈ I ε i ( b )Λ i and ϕ ( b ) = (cid:88) i ∈ I ϕ i ( b )Λ i . The beauty about crystal theory is that it is well-behaved with respect to taking tensor products.Let B and B be two g -crystals. As a set B ⊗ B is the Cartesian product of the two sets.For b = b ⊗ b ∈ B ⊗ B , the weight function is simply wt( b ) = wt( b ) + wt( b ). The crystaloperators are given by f i ( b ⊗ b ) = (cid:40) f i ( b ) ⊗ b if ε i ( b ) ≥ ϕ i ( b ), b ⊗ f i ( b ) otherwise,and similarly for e i ( b ). This rule can also be expressed combinatorially by the signature rule.A highest weight crystal B ( λ ) of highest weight λ ∈ P + is a crystal with a unique element u λ such that e i ( u λ ) = ∅ for all i ∈ I and wt( u λ ) = λ . On finite-dimensional highest weightcrystals B ( λ ) there exists an involution S : B ( λ ) → B ( λ ), called the Lusztig involution , whichis a crystal isomorphism such that S ( f i ) = e i ∗ and S ( e i ) = f i ∗ . Here i ∗ is defined through the map α i (cid:55)→ α i ∗ := − w ( α i ) with w the longest element in theWeyl group of g . Explicitly, we have i ∗ = n − i for type A n − and i ∗ = i for type C n . Under S the highest weight element goes to the lowest weight element.In [8], Henriques and Kamnitzer defined a crystal commutor on the tensor product of twoclassically highest weight crystals in terms of Lusztig’s involution as B ( λ ) ⊗ B ( µ ) → B ( µ ) ⊗ B ( λ ) b ⊗ b (cid:55)→ S ( S ( b ) ⊗ S ( b )) . (2.1) RYSTAL ENERGY VIA CHARGE 5
Kashiwara–Nakashima columns for type C . Kashiwara and Nakashima [17] de-veloped general tableaux models for finite-dimensional highest weight crystals for all non-exceptional classical Lie algebras g . For type C n , the Kashiwara–Nakashima (KN) columns[17] of height k index the vertices of the fundamental representation V ( ω k ) of the symplecticalgebra sp n ( C ). These columns are filled with entries in [ n ] := { < < · · · < n < n < n − < · · · < } . Definition 2.1.
A column-strict filling b = b (1) . . . b ( k ) with entries in [ n ] is a KN column ifthere is no pair ( z, z ) of letters in b such that: z = b ( p ) , z = b ( q ) , q − p ≤ k − z . We will need a different definition of KN columns, which was proved to be equivalent to theone above in [44].
Definition 2.2.
Let b be a column and I = { z > · · · > z r } the set of unbarred letters z suchthat the pair ( z, z ) occurs in b . The column b can be split when there exists a set of r unbarredletters J = { t > · · · > t r } ⊂ [ n ] such that: • t is the greatest letter in [ n ] satisfying: t < z , t (cid:54)∈ b , and t (cid:54)∈ b , • for i = 2 , . . . , r , the letter t i is the greatest one in [ n ] satisfying t i < min( t i − , z i ) , t i (cid:54)∈ b ,and t i (cid:54)∈ b .In this case we write: • b R for the column obtained by changing z i into t i in b for each letter z i ∈ I , and byreordering if necessary, • b L for the column obtained by changing z i into t i in b for each letter z i ∈ I , and byreordering if necessary.The pair b L b R will be called a split column. Example 2.3.
The following is a KN column of height 5 in type C n for n ≥
5, together withthe corresponding split column: b = 45543 , b L b R = 1 42 55 34 23 1 . We used the fact that { z > z } = { > } , so { t > t } = { > } .We will consider Definition 2.2 as the definition of KN columns.2.3. Kirillov–Reshetikhin crystals.
For the definition of the crystal energy function, weneed to endow the KN columns with an affine crystal structure. These finite-dimensionalaffine crystals are called
Kirillov–Reshetikhin (KR) crystals . Combinatorial models for all non-exceptional types were provided in [5].Here we only describe the KR crystals B r, for types A (1) n − and C (1) n , where r ∈ { , , . . . , n − } and r ∈ { , , . . . , n } , respectively. As a classical type A n − (resp. C n ) crystal, the KR crystalis isomorphic to B r, ∼ = B ( ω r ) . C. LENART AND A. SCHILLING
The crystal operator f is given as follows. Let b ∈ B k, , represented by a one-columnKN tableau. In type A n − , if b contains the letter n and no 1, f ( b ) is the obtained from b by removing n and adding 1 to the column, leaving all letters in strictly increasing order.Otherwise f ( b ) = ∅ . In type C n , if b contains the letter 1, then f ( b ) is obtained from b byremoving the 1 and adding the letter 1, arranging all letters again in strictly increasing order.Otherwise f ( b ) = ∅ . Note that if b contains 1, then it cannot contain 1 by the KN conditionof Definition 2.1.Similarly, in type A n − , e ( b ) changes a 1 into n if 1 is in b and n is not, and otherwise e ( b ) = ∅ . In type C n , e ( b ) is obtained from b by changing a 1 into a 1 if it exists, and e ( b ) = ∅ otherwise.2.4. The D function. Now we come to the definition of the energy function D on tensorproducts of KR crystals B r, of type A (1) n − or C (1) n . It is defined by summing up combinatoriallydefined “local” energy contributions using the combinatorial R -matrix.Let B , B be two affine crystals with generators v and v , respectively, such that B ⊗ B is connected and v ⊗ v lies in a one-dimensional weight space. By [28, Proposition 3.8], thisholds for any two KR crystals. The generator v r,s for the KR crystal B r,s is the unique elementof classical weight sω r .The combinatorial R -matrix [14, Section 4] is the unique crystal isomorphism σ : B ⊗ B → B ⊗ B . By weight considerations, this must satisfy σ ( v ⊗ v ) = v ⊗ v .As in [14] and [39, Theorem 2.4], there is a function H = H B ,B : B ⊗ B → Z , unique upto a global additive constant, such that, for all b ∈ B and b ∈ B ,(2.2) H ( e i ( b ⊗ b )) = H ( b ⊗ b ) + − i = 0 and LL,1 if i = 0 and RR,0 otherwise.Here LL (resp. RR) indicates that e acts on the left (resp. right) tensor factor in both b ⊗ b and σ ( b ⊗ b ). When B and B are KR crystals, we normalize H B ,B by requiring H B ,B ( v ⊗ v ) = 0, where v and v are the generators defined above. The map H is calledthe local energy function . Definition 2.4.
For B = B r N , ⊗ · · · ⊗ B r , of type A (1) n − or C (1) n , set H Rj,i := H i σ i +1 σ i +2 · · · σ j − and H Lj,i := H j − σ j − σ j − · · · σ i , where σ j and H j act on the j -th and ( j + 1) -st tensor factors. We define a right and left energyfunction D RB , D LB : B → Z as (2.3) D RB := (cid:88) N ≥ j>i ≥ H Rj,i and D LB := (cid:88) N ≥ j>i ≥ H Lj,i . We set D B := D LB and, when there is no confusion, we shorten D B to simply D ; this is referredto as the energy function.Remark . Note that the energies D RB and D LB can be defined for general tensor products ofKR crystals. When the KR crystals decompose into several components as classical crystals RYSTAL ENERGY VIA CHARGE 7 (unlike in the cases for type A (1) n − and C (1) n we consider), there is an extra term in the analogueof (2.3); see [39].There is a precise relationship between D R and D L using the Lusztig involution. To stateit, let us introduce the following map τ : B N ⊗ · · · ⊗ B → B ⊗ · · · ⊗ B N b N ⊗ · · · ⊗ b (cid:55)→ S ( b ) ⊗ · · · ⊗ S ( b N ) . For types A (1) n − and C (1) n and B i = B r i , , the KR crystal B i is connected as a classical crystaland under S the classically highest weight element u highest i maps to the lowest weight element u lowest i . It is not hard to show from the explicit description of S , e and f in this case, thatthe following diagrams commute:(2.4) B r, f (cid:47) (cid:47) S (cid:15) (cid:15) B r, S (cid:15) (cid:15) B r, e (cid:47) (cid:47) B r, and B f (cid:47) (cid:47) τ (cid:15) (cid:15) B τ (cid:15) (cid:15) (cid:101) B e (cid:47) (cid:47) (cid:101) B where B = B r N , ⊗ · · · ⊗ B r , and (cid:101) B = B r , ⊗ · · · ⊗ B r N , .This shows in particular that the crystal commutor (2.1) is lifted to an affine crystal isomor-phism in these cases and hence must coincide with the combinatorial R -matrix σ . Proposition 2.6.
Let B = B r N , ⊗ · · · ⊗ B r , of type A (1) n − or C (1) n and b ∈ B . Then (2.5) D RB ( b ) = D LB ( τ ( b )) . Proof.
Note that if σ ( b ⊗ b ) = b (cid:48) ⊗ b (cid:48) , then σ ( S ( b ) ⊗ S ( b )) = S ( b (cid:48) ) ⊗ S ( b (cid:48) ) since σ andthe commutor (2.1) agree on two tensor factors under the conditions of the proposition by theabove arguments. Hence the terms in the sum of D R are in one-to-one correspondence withterms in D L . Therefore it suffices to show that the local energy satisfies(2.6) H ( b ⊗ b ) = H ( S ( b ) ⊗ S ( b ))for b ⊗ b ∈ B r , ⊗ B r , . Since S ( u highest1 ) ⊗ S ( u highest2 ) = u lowest1 ⊗ u lowest2 lies in the samecomponent as u highest1 ⊗ u highest2 , we have H ( u highest2 ⊗ u highest1 ) = H ( S ( u highest1 ) ⊗ S ( u highest2 )) = 0 . In the recursion (2.2), if for example e acts LL on b ⊗ b , then f acts RR on S ( b ) ⊗ S ( b ) = S ( b ) ⊗ S ( b ) by (2.4). Hence H changes by the same amount in the left and right hand sideof (2.6). The other cases can be checked analogously, which proves (2.6). (cid:3) D energy as affine grading. As suggested in [38, Section 2.5] and proven in [42], theenergy D R is the same as the affine degree grading in the associated highest weight affinecrystals up to an overall shift. We will explain this now since it plays a crucial role in the proofof the equality between charge and energy.We begin with the definition of Demazure arrows. For this we need constants c r for r ∈ I \{ } as for example defined in [6]. In the cases of concern to us here, we have c r = 1 for all r in type A (1) n − and c r = 2 for 1 ≤ r < n and c n = 1 in type C (1) n . C. LENART AND A. SCHILLING
Definition 2.7.
Let B = B r N ,s N ⊗ · · · ⊗ B r ,s be a tensor product of KR crystals and fix aninteger (cid:96) such that (cid:96) ≥ (cid:100) s k /c k (cid:101) for all ≤ k ≤ N . We call such a tensor product a compositeKR crystal of level bounded by (cid:96) .An arrow f i is called an (cid:96) -Demazure arrow on b ∈ B if ϕ i ( b ) > and either i ∈ I \ { } or i = 0 and ε ( b ) ≥ (cid:96) . In the setting of this paper, we are only concerned with tensor products of types A (1) n − and C (1) n of the form B = B r N , ⊗ · · · ⊗ B r , . In this case one can pick (cid:96) = 1 and a Demazure arrow for B is a 1-Demazure arrow. Lemma 2.8.
Let B = B r N , ⊗ · · · ⊗ B r , of type A (1) n − or C (1) n and b ∈ B . Then (1) ε ( b ) ≥ implies D R ( f ( b )) = D R ( b ) + 1 ; (2) ϕ ( b ) ≥ implies D L ( e ( b )) = D L ( b ) + 1 .Proof. Part (1) follows directly from [42, Lemma 7.3]. For part (2), recall that by (2.4) andProposition 2.6 we have e ( τ ( b )) = τ ( f ( b )) and D L ( τ ( b )) = D R ( b ). Also, setting ˜ b = τ ( b ), wehave ϕ (˜ b ) ≥ ε ( b ) ≥
1; thus, by using part (1), we deduce D L ( e (˜ b )) = D L ( e ( τ ( b ))) = D L ( τ ( f ( b ))) = D R ( f ( b )) = D R ( b ) + 1 = D L (˜ b ) + 1 , which proves the claim. (cid:3) The proof of the following essentially appeared in [14, Proof of Theorem 4.4.1] and wasspelled out in this precise form in [42, Proposition 8.1]. Here P + (cid:96) = { λ ∈ P + | lev( λ ) = (cid:96) } , where lev( λ ) := λ ( c ) is the level of λ and c is the central element c = (cid:80) i ∈ I a ∨ i α ∨ i . Proposition 2.9.
For B a composite KR crystal of level bounded by (cid:96) , (2.7) B ⊗ B ( (cid:96) Λ ) ∼ = (cid:77) Λ (cid:48) B (Λ (cid:48) ) , where the sum is over a finite collection of (not necessarily distinct) Λ (cid:48) ∈ P + (cid:96) . In Section 5, we discuss in more detail the sum on the right hand side of (2.7) for type C (1) n and (cid:96) = 1. Definition 2.10.
For each b ∈ B , let u (cid:96) Λ b be the unique element of B such that u (cid:96) Λ b ⊗ u (cid:96) Λ isthe highest weight in the component from Proposition containing b ⊗ u (cid:96) Λ . Define the function deg on a direct sum of highest weight crystals to be the basic grading oneach component, with all highest weight elements placed in degree 0.
Corollary 2.11.
Choose an isomorphism m : B ⊗ B ( (cid:96) Λ ) ∼ = (cid:76) Λ (cid:48) B (Λ (cid:48) ) . Then for all b ∈ B ,we have D ( b ) − D ( u (cid:96) Λ b ) = deg( m ( b ⊗ u (cid:96) Λ )) . Corollary 2.12.
The minimal number of e in a string of e i taking b to u (cid:96) Λ b is D ( b ) − D ( u (cid:96) Λ b ) .Remark . One special case of interest is when B is a tensor product of perfect KR crystalsof level (cid:96) , and Λ also has level exactly (cid:96) . Then the right side of (2.7) is a single highest weightcrystal and the isomorphism in Proposition 2.9 is used in the Kyoto path model [14]. Hence u (cid:96) Λ b does not in fact depend on b , simplifying Corollaries 2.11 and 2.12. RYSTAL ENERGY VIA CHARGE 9 The charge construction
The classical charge.
Let us start by recalling the construction of the classical chargeof a word due to Lascoux and Sch¨utzenberger [20]. Assume that w is a word with letters in thealphabet [ n ] := { , . . . , n } which has partition content, i.e., the number of j ’s is greater than orequal to the number of j + 1’s, for each j = 1 , . . . , n −
1. The statistic charge( w ) is calculatedas a sum based on the following algorithm. Scan the word starting from its right end, andselect the numbers 1 , , . . . in this order, up to the largest possible k . We always pick the firstavailable entry j + 1 to the left of the previous entry j . Whenever there is no such entry, wepick the rightmost entry j + 1, so we start scanning the word from its right end once again; inthis case, we also add k − j to the sum that computes charge( w ). At the end of this process, weremove the selected numbers and repeat the whole procedure until the word becomes empty. Example 3.1.
Consider the word w = 11
3, so thecontribution to the charge is 2. We are left with the word 123, whose contribution to the chargeis 2 + 1 = 3. So charge( w ) = 1 + 2 + (2 + 1) = 6.We now reinterpret the classical charge as a statistic on a tensor product of type A (1) n − KRcrystals. Such a crystal indexed by a column of height k is traditionally denoted B k, , and itsvertices are indexed by increasing fillings of the mentioned column with integers in [ n ]. Givena partition µ (i.e., a dominant weight in the root system), let(3.1) B µ := µ (cid:79) i =1 B µ (cid:48) i , , where µ (cid:48) is the conjugate partition to µ . This is simply the set of column-strict fillings of theYoung diagram µ with integers in [ n ]. Note that unlike in Section 2, the tensor factors in (3.1)are ordered in weakly decreasing order.Fix a filling b in B µ written as a concatenation of columns b . . . b µ . We attach to it a filling c := circ-ord( b ) = c . . . c µ according to the following algorithm, which is based on the circularorder ≺ i on [ n ] starting at i , namely i ≺ i i + 1 ≺ i · · · ≺ i n ≺ i ≺ i · · · ≺ i i − Algorithm 3.2. let c := b ;for j from to µ dofor i from to µ (cid:48) j dolet c j ( i ) := min ( b j \ { c j (1) , . . . , c j ( i − } , ≺ c j − ( i ) ) end do;end do;return c := c . . . c µ . Example 3.3.
Algorithm 3.2 constructs the filling c from the filling b below. The bold entriesin c are only relevant in Example 3.5 below.(3.2) b = 3 2 1 25 3 26 4 4 and c = 3 3 . We introduce some terminology in order to reinterpret the classical charge in terms of astatistic on B µ . Given the considered filling b in B µ , we define its charge word as the biwordcw( b ) containing a biletter (cid:0) kj (cid:1) for each entry k in the column b j of b . We order the biletters inthe decreasing order of the k ’s, and for equal k ’s, in the decreasing order of j ’s. The obtainedword formed by the lower letters j will be denoted by cw ( b ). We refer to Example 3.5 for anillustration of the charge word. On the other hand, given the filling c = c . . . c µ constructedby Algorithm 3.2, we say that the cell γ in column c j and row i is a descent if c j ( i ) > c j +1 ( i ),assuming that c j +1 ( i ) is defined. Let Des( c ) denote the set of descents in c . As usual, we definethe arm length arm( γ ) of a cell γ as the number of cells to its right.It is not hard to see that Algorithm 3.2 for constructing c from b translates precisely into theselection algorithm which computes charge(cw ( b )). More precisely, consider the i th sequence1 , , . . . extracted from cw ( b ) (which turns out to have length µ i ), and the letter j in thissequence; then the top letter paired with the mentioned letter j in cw( b ) is precisely the entry c j ( i ) in row i and column j of the filling c . In particular, the steps to the right in the i thiteration of the charge computation correspond precisely to the descents in the i th row of c ,while the corresponding charge contributions and arm lengths coincide. We conclude that(3.3) (cid:88) γ ∈ Des( c ) arm( γ ) = charge(cw ( b )) . For simplicity, we set charge( b ) := charge(cw ( b )). Remark . In [30] we showed that the charge statistic on B µ can be derived from the Ram–Yip formula [40] for the corresponding Macdonald polynomial at t = 0. In fact, we showed thatAlgorithm 3.2 is closely related to the corresponding quantum Bruhat graph (see, e.g., [2]). Sowe can conclude that this graph explains the charge construction itself. The mentioned ideawas extended to type C , and it led to the definition of a type C charge, that we describe inSection 3.2. Example 3.5.
Note that cw ( b ) for b in Example 3.3 is precisely the word w in Example 3.1.In fact, the full biword cw( b ) is shown below, using the order on the biletters specified above.The index attached to a lower letter is the number of the iteration in which the given letter isselected in the process of computing charge( b ):cw( b ) = (cid:18) (cid:19) . One can note the parallel between the mentioned selection process and the construction of c from b in Example 3.3. The entries in the cells of Des( c ) are shown in bold in (3.2).3.2. The type C charge. In this section we recall from [30] the construction of the type C charge. We start by fixing a dominant weight µ in the root system of type C n . Let(3.4) B µ := µ (cid:79) i =1 B µ (cid:48) i , , where B k, is the type C (1) n KR crystal indexed by a column of height k . Note that B µ is the setof fillings b = b . . . b µ of the shape µ with integers in [ n ] whose columns b j are KN columns;indeed, the KN columns of height k label the vertices of B k, . As mentioned above, it will bemore useful to represent b j in the split form b Lj b Rj ; in this case, b becomes a filling b L b R . . . b Lµ b Rµ of the shape 2 µ . RYSTAL ENERGY VIA CHARGE 11
Now fix a filling b in B µ , represented with split columns, which are labeled from left to rightby 1 , (cid:48) , , (cid:48) , . . . . We can apply a slight modification of Algorithm 3.2 to b and obtain a filling c = c L c R . . . c Lµ c Rµ = circ-ord( b ) of 2 µ ; namely, we start by setting c L := b L , and then considerthe (doubled) columns of b from left to right. We use the circular order on [ n ] starting at variousvalues i , which we still denote by ≺ i . Example 3.6.
Consider the following tensor product of KN columns:5321 ⊗ ⊗ . This is represented with split columns as the following filling b of the shape 2 µ = (6 , , , (cid:48) (cid:48) (cid:48) , where the top row consists of the column labels. The corresponding filling c is5 5 4 4 3 23 3 3 . Define the charge word cw( b ) of b by analogy with type A , as the biword containing a biletter (cid:0) kj (cid:1) for each entry k in column j of b ; here j and k belong to the alphabets { < (cid:48) < < (cid:48) < . . . } and [ n ], respectively. We order the biletters as in the type A case (in the decreasing order ofthe k ’s, and for equal k ’s, in the decreasing order of j ’s), and define cw ( b ) in the same way (asthe word formed by the lower letters j ).The modification of Algorithm 3.2 for constructing c from b can be rephrased in terms ofcw( b ), as explained below; we will refer to this rephrasing as the charge algorithm. We startby scanning cw ( b ) from right to left and by selecting the entries 1 , (cid:48) , , (cid:48) , . . . , µ , ( µ ) (cid:48) in thisorder, according to the following rule: always pick the first available entry to the left, but if thedesired entry is not available then scan the word from its right end once again. As in type A ,we can see that the sequence of top letters paired with 1 , (cid:48) , , (cid:48) , . . . , µ , ( µ ) (cid:48) is the first rowof the filling c (read from right to left). We then remove the selected entries from cw ( b ) andrepeat the above procedure, which will now give the other rows of c , from top to bottom. Itwas shown in [30] that we always go left from j to j (cid:48) , but we can go right from j (cid:48) to j + 1. Example 3.7.
This is a continuation of Example 3.6. The charge word cw( b ), with the orderon the biletters indicated above, is (cid:18) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) (cid:19) . The index attached to a lower letter is the number of the iteration in which the given letter isselected by the charge algorithm.Descents are defined as usual, cf. Section 3.1. It is easy to see that the descents in c correspondto the steps to the right in the charge algorithm applied to cw ( b ). By an observation made above, we only have descents of the form c Rj ( i ) > c Lj +1 ( i ). We are led to the following definitionof the type C charge. Definition 3.8.
Consider a word w with letters in the alphabet , (cid:48) , , (cid:48) , . . . , containing asmany letters j as j (cid:48) , and at least as many letters j as j + 1 . Apply the charge algorithm to w , and assume that a selected entry j (cid:48) is always to the left of the previously selected j . Let charge( w ) be the sum of k − j for each selected entry j + 1 to the right of the previously selected j (cid:48) , where the selected entries in the given iteration are , (cid:48) , . . . , k, k (cid:48) . The above discussion leads to the following result:(3.5) 12 (cid:88) γ ∈ Des( c ) arm( γ ) = charge(cw ( b )) . For simplicity, we again set charge( b ) := charge(cw ( b )). Example 3.9.
This is still a continuation of Example 3.6. The entries in the descents of c are shown above in bold. Correspondingly, the charge algorithm applied to cw ( b ) makes onestep to the right in the second iteration (from 2 (cid:48) to 3), and two steps to the right in the thirditeration (from 1 (cid:48) to 2 and from 2 (cid:48) to 3). Thus, charge( b ) = 1 + (2 + 1) = 4.4. Energy and charge in type A In this section we rederive the result of Nakayashiki–Yamada [37] showing the equality ofthe energy function and charge in type A n − . We do this in a more conceptual way, by usingthe method of Demazure arrows (the proof in [37] is based on subtle combinatorics of Youngtableaux). Furthermore, we work with all the crystal vertices in a tensor product of columns,not just the highest weight vertices considered in [37]. Note that the setup in the mentionedpaper is that of the right energy and a tensor product of columns of increasing heights which,by Proposition 2.6, is equivalent to the setup in this paper.We start by studying the behavior of the type A charge with respect to the crystal operators. Proposition 4.1.
The type A n − charge is preserved by the crystal operators f , . . . , f n − .Proof. Let b be a tensor product of columns in some B µ (see Section 3.1 and the terminologytherein, which we use freely). It is known that the word cw ( f i ( b )) is in the same placticequivalence class (see, e.g., [21]) as cw ( b ). More precisely, the former is obtained from thelatter by considering its subword formed by the letters x corresponding to biletters (cid:18) i + 1 x (cid:19) or (cid:18) ix (cid:19) in cw( b ), by viewing this subword as (the column word of) a skew tableau with twocolumns, and by using jeu de taquin to move a letter from the right column to the left one.This is explained in detail in [22, Section 2], based on the notion of “double crystal graphs”.Then we use the well-known fact that the classical charge is preserved by the plactic relations(see, e.g., [21, Lemma 6.6.6 (ii)]). (cid:3) Proposition 4.2.
Let B = B r N , ⊗ · · · ⊗ B r , be of type A (1) n − with r N ≥ r N − ≥ · · · ≥ r and b ∈ B . If ϕ ( b ) ≥ and ε ( b ) ≥ , then the type A n − charge satisfies charge( e ( b )) = charge( b ) − . RYSTAL ENERGY VIA CHARGE 13
Proof.
Let b = b . . . b µ , and assume that e changes the entry 1 in b j to n . The condition ϕ ( b ) ≥ j >
1. Let c := circ-ord( b ) with c = c . . . c µ , where c j ( i ) = 1, and d := circ-ord( e ( b )) with d = d . . . d µ (recall that the map circ-ord is defined by Algorithm 3.2). The main case.
We start by assuming that c j − ( i ) > c j +1 , if it exists,does not contain n in a row k ≥ i . We claim that, in this case, d l ( k ) = c l ( k ) for all k, l withthe exception of d j ( i ) = n . Indeed, all we need to check are the following, in this order: (1) theentries d j ( k ) for k = 1 , . . . , i − n , would lead to a contradiction; (2) the value of d j ( i ) follows from the fact that c j − ( i ) (cid:54) = 1;(3) the value of d j +1 ( i ), if this entry exists, follows from the above condition on the column c j +1 . By the above facts, we have the same descents in the fillings c and d , with the exceptionof the descent c j − ( i ) > c j ( i ) = 1, which corresponds to d j ( i ) = n > d j +1 ( i ), assuming that d j +1 ( i ) exists. The difference between the arm lengths of the mentioned descents is 1, whichconcludes the proof by (3.3). The remaining part of the proof treats the exceptions to this case. Exception . Assume that c j − ( i ) = 1. Then the column b j − must contain n (otherwise theentry 1 in b j would not be the leftmost unpaired 1). Assuming that c j − ( k ) = n , we must have k > i , by Algorithm 3.2 and the fact that c j ( i ) = 1. The condition ϕ ( b ) ≥ j >
2, andwe have c j − ( i ) = 1 (otherwise c j − ( i ) = 1 and c j − ( k ) = n for k > i are in contradiction withthe way in which Algorithm 3.2 reorders the column b j − ). This reasoning can be continuedindefinitely, so this case is impossible. Exception . Assume that c j − ( i ) >
1, but c j +1 ( k ) = n for some k ≥ i . Then b j +1 mustcontain 1 as well, namely c j +1 ( l ) = 1 (otherwise the entry 1 in b j would be paired with theentry n in b j +1 ). We must have l ≤ i , by Algorithm 3.2 and the fact that c j ( i ) = 1. The abovefacts imply that l < k . Exception . Assume that l < i . The fact that c j ( l ) > c j +1 ( l ) = 1, and c j +1 ( k ) = n for k > l are in contradiction with the way in which Algorithm 3.2 reorders the column b j +1 . Sothis case is impossible. Exception . The only possibility left is that c j +1 ( i ) = 1 and c j +1 ( k ) = n for k = k > i .Let us assume for the moment that the second condition in Exception 2 holds for the column c j +2 , namely c j +2 ( k ) = n for some k ≥ i . By the same reasoning as above, we deduce that c j +2 ( i ) = 1, so k > i . In fact, we also have k ≥ k , by Algorithm 3.2. By continuing thisreasoning, we obtain c j +1 ( i ) = . . . = c j + p ( i ) = 1 , c j +1 ( k ) = . . . = c j + p ( k p ) = n , where k ≥ . . . ≥ k p ;on the other hand, we can assume that, if the column c j + p +1 exists, then it does not contain n in rows i, i + 1 , . . . . This information about the filling c is represented in the figure below.The column c j is the leftmost column with an entry 1 displayed, while + and ∗ stand forentries different from 1 and n , respectively. The boxes shown in bold represent descents. By areasoning similar to the main case above, we deduce that the only difference between the fillings c and d consists of the entries 1 and n in the figure changing to n and 1 in d , respectively. Thisleads to the marked descents moving to the boxes indicated by the arrows. It is now easy tosee that the sum of the arm lengths of descents decreases by 1 when passing from c to d . ... ++++ n n * * n n *n n * ......... ... ...... ...... ... (cid:3) Proof of Theorem in type A . The proof is immediate based on Corollary 2.11 and Proposi-tions 4.1, 4.2 using the fact that KR crystals of type A (1) n − are perfect. (cid:3) Kyoto path model for nonperfect type C In this section, we make Proposition 2.9 more explicit in the case of B = B r N , ⊗ · · · ⊗ B r , and (cid:96) = 1 for type C (1) n , by providing a correspondence between highest weight elements (orground states) in B ⊗ B (Λ ) and elements in B (Λ (cid:48) ) in the sum on the right hand side of (2.7),which are of type A . This will help in the next section to prove Theorem 1.1 for type C (1) n .We call the highest weight elements in B ⊗ B (Λ ) ground state paths . There is a recursiveconstruction for them, which starts by listing all elements b ∈ B r , such that ε ( b ) = Λ .Suppose b k ⊗ · · · ⊗ b ∈ B r k , ⊗ · · · ⊗ B r , are already constructed. Then b k +1 ∈ B r k +1 , can beany of the elements such that ε ( b k +1 ) = ϕ ( b k ). The weight of the ground state is ϕ ( b N ), whichis some fundamental weight Λ h . For perfect crystals there are unique elements b N , . . . , b withthe described properties. However, in type C (1) n the crystals B r, are not perfect and the aboveconstruction gives a tree of ground state elements. Example 5.1.
Take B = B , ⊗ B , ⊗ B , ⊗ B , of type C (1)3 . Then b is the column 321and b the column 23. For b there are two choices, namely the columns 32 or 22. In the firstcase b is 3, and in the second case b can be 2 or 1. In summary the three ground states are3 ⊗ ⊗ ⊗ ⊗ u Λ ⊗ ⊗ ⊗ ⊗ u Λ ⊗ ⊗ ⊗ ⊗ u Λ . The weights are Λ , Λ , and Λ , respectively. Theorem 5.2.
Let B = B r N , ⊗ · · · ⊗ B r , of type C (1) n . From each ground state u ⊗ u Λ ∈ B ⊗ B (Λ ) there exists a sequence of Demazure arrows f i (see Definition ), which ends atan element b ⊗ u Λ such that b does not contain any barred letter. Note that in Theorem 5.2 there is no assumption on the order of the r i . RYSTAL ENERGY VIA CHARGE 15
In order to provide a proof of Theorem 5.2, we describe the explicit sequence of f i satisfyingthe required conditions. For this we recursively define the following objects.- Let λ ⊆ λ ⊆ · · · ⊆ λ s ( s is given below) be a sequence of shapes, where λ is a singlecolumn of height h if the weight of the ground state path u ⊗ u Λ is Λ h . The othershapes are all of the form λ ( k, h , h ), a partition with k columns of height n followedby two columns of heights n > h ≥ h ≥
0, respectively. If λ j = λ ( k, h , h ), then λ j +1 = λ ( k, h + 1 , h + 1), where we identify λ ( k, h + 1 , h + 1) ∼ = λ ( k + 1 , h + 1 , h + 1 = n and h + 1 < n , and λ ( k + 2 , ,
0) if h + 1 = h + 1 = n . This adds onehorizontal domino in the consecutive rightmost columns up to height n .- We recursively define v j +1 := F j ( v j )with v = u the ground state and F j the sequence F j := f k +10 f k +21 · · · f k +2 h f k +1 h +1 · · · f k +1 h f kh +1 · · · f kn − f kn , if λ j = λ ( k, h , h ). We continue doing this as long as possible, in other words, until F s ( v s ) is undefined. Example 5.3.
Let u ⊗ u Λ be the second ground state from Example 5.1. Then the sequenceof λ ⊆ λ ⊆ · · · ⊆ λ s is ⊆ ⊆ ⊆ with F = f f f F = f f f f F = f f f f . Let λ j = ( λ j ≥ λ j ≥ . . . ), and consider the conjugate partition λ (cid:48) j = ( λ (cid:48) j ≥ λ (cid:48) j ≥ . . . ). Wewill now explain how to represent an element v j by a collection of non-crossing lattice paths(which might touch each other); see Figure 1. The paths have the following types of steps: down(southeast), up (northeast), and horizontal (east). More precisely, the paths in the collection P j = { p , . . . , p l } representing v j correspond to the l = λ j columns of λ j , and they satisfy theconditions below.(1) The endpoints of p , . . . , p l are aligned at height 0 from right to left, and p i starts atheight λ (cid:48) ji .(2) The paths p , . . . , p l and the segment of p before the end of p (in case p exists)consist entirely of down and horizontal steps. Two up or down steps never lie belowone another, and neither do only horizontal steps.(3) The path p i starts after p i +2 ends, for i = 1 , . . . , l − x -coordinates, withequality allowed. Each down (resp. up) step starting at height i is labeled by a letter i (resp.¯ ı ). We define word( P j ) as the word obtained by reading the labels on the paths in P j from leftto right (recall that two labels never lie below one another).Let us now explain the construction of the collections P j . We start by defining P as consistingof a single path: the one having the same word as u = v , ending at height 0, and having nohorizontal steps; the word of u , denoted word( u ), is defined as usual, by reading its columns from left to right, bottom to top. We construct the collections P j recursively, via the followingtransformation rule P (cid:55)→ up( P ) on a collection of paths P = ( p , . . . , p l ) satisfying conditions(1)-(3) above. Rule 5.4. - Locate the leftmost up step in p , and let y be its label. Replace it with a horizontal stepat height y and a down step below it ending at height . - Shift up by the segment of p to the left of the position where the above change occurred,to connect it to the tail of p . Shift up by all the other paths. - Replace any down step above height n by a down step below it, ending at height . - Consider the down steps ending at height from right to left, excluding the rightmostone. Match them with the shifts of p , . . . , p l , in this order, and connect the matchedpairs by horizontal lines of height . Note that, in the last step of the rule, the last one or two down steps might have no match,so they start new paths. For simplicity, any horizontal steps at the beginning of a path areignored. It is easy to see that, since P satisfies conditions (1)-(3) above, the rule can be applied,and up( P ) satisfies the same conditions. Thus, we can recursively define P j +1 := up( P j ) aslong as there are up steps in P j .We claim that applying the rule to P j corresponds to applying F j to v j . To make this precise,we introduce some notation. Let r be the collection of columns of B = B r N , ⊗ · · · ⊗ B r , , i.e.,the tuple ( r N , . . . , r ). Given a word w of length r + · · · + r N , we define a rearrangement ofit ord r ( w ) by slicing w into segments of length r N , . . . , r in this order, and by reordering eachsegment decreasingly. We will show below that, for all j = 0 , . . . , s , we have(5.1) ord r (word( P j )) = word( v j ) . We will see that the reordering of the segments of word( P j ) done by ord r , if any, is verysimple: a segment is a concatenation of two decreasing segments which can be swapped to givea decreasing sequence. Based on the above discussion, one can see that λ j is the weight of v j ,and that this element is of highest weight in its classical (non-affine) component. Example 5.5.
The collections of paths P j corresponding to Example 5.3 is shown in Figure 1. Proof of Theorem . It essentially suffices to show that F j ( v j ) is defined if and only if up( P j ) isdefined and that (5.1) holds, for all j (the only extra fact to check is that all arrows correspondingto F j are Demazure arrows, see below). Indeed, Rule 5.4 can be applied as long as there areup steps in P j , so the final element v s will have only positive entries. We will first show thatRule 5.4 precisely describes the action of F j on the word of P j , viewed as an element of thetensor product ( B , ) ⊗ ( r + ··· + r N ) ; more precisely, we have(5.2) F j (word( P j )) = word( P j +1 ) . We will then show that the above action of F j is completely similar to that on v j , which willprove (5.1).Let us describe the action of F j on word( P j ). First note that the connected components of thepaths in P j correspond to bracketed units in the tensor product ( B , ) ⊗ ( r + ··· + r N ) according tothe signature rule for the application of the Kashiwara operator on tensor products of crystals.This means that any given operator f i can only operate on the starting points of the connectedpaths in P j . The application of f kn lifts all starting points of value n to n . Note that by the RYSTAL ENERGY VIA CHARGE 17 v = 2 ⊗ ⊗ ⊗ (cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:0)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:0)(cid:0)(cid:64)(cid:64) v = 3 ⊗ ⊗ ⊗ (cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:0)(cid:0)(cid:0)(cid:0)(cid:64)(cid:64)(cid:64)(cid:64) (cid:64)(cid:64) v = 1 ⊗ ⊗ ⊗ (cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:0)(cid:0)(cid:64)(cid:64) (cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) v = 2 ⊗ ⊗ ⊗ (cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) (cid:64)(cid:64)(cid:64)(cid:64) Figure 1.
Paths for Example 5.5weight λ j = λ ( k, h , h ) of v j there are precisely k of them. Next f kn − lifts all k just created n to n − k letters n − n that were previouslybracketed with the just lifted n etc.. Potentially there are two more connected path componentswith starting points at h and h , hence the exponents of f i with i ≤ h and i ≤ h are increasedaccordingly. In summary, the starting points n of path components are eventually lifted to 1and then turned into 1 by f ; the other positive letters to the left of the leftmost up step are allraised by one. In the rightmost path, this lifting process eventually reaches the leftmost up step y which is lifted to y − f y − . The steps to the right of the leftmost up step y are either y + 1 or y . Both would need an f y to be lifted. Since the f i in F j are applied in decreasingorder of the indices i , this cannot happen and hence the leftmost up step is lifted to 1 and thenturned into 1 by f . Altogether, the changes are precisely as described in Rule 5.4. Let us now compare the action of F j on word( P j ), i.e., on ( B , ) ⊗ ( r + ··· + r N ) , with that on v j ,i.e., on B = B r N , ⊗ · · · ⊗ B r , . Note that by the action of f on columns rather than tensorproducts of single boxes, the word word( P j ) of a path is a concatenation of cyclically shiftedcolumns of v j . We claim that F j acts on precisely the same entries in the two cases. In theexpression for F j the f i with larger indices i act first and hence lift up the upper or barredportion of the paths. This is the case before or after reordering. f changes a ¯1 into a 1 in bothcases. Hence the action commutes with the reordering.Finally, the rightmost tensor factor in v is the column c = r · · · ∈ B r , . Note that thiscolumn is never changed during the algorithm and satisfies ε ( c ) = 1. By the tensor productrules this implies that ε ( v j ) ≥ ≤ j < s , so that all arrows are Demazure arrows. (cid:3) There is a more direct way of constructing the type A elements b in Theorem 5.2 from theground state u ⊗ u Λ . For i = 1 , , . . . , N , place the letter i in column 1 for each unbarred letterin u i and in column 2 for each barred letter, where u = u N ⊗ · · · ⊗ u . Note that, due to thefact that u is a ground state, the difference in height between the first and second column isat most n . Now cut the columns at heights n, n, n, . . . and put all pieces next to each otheraligned at height 0. The letters in row i record the tensor factors of b in Theorem 5.2 whichcontain the letter i . Comparing this with the algorithm in the proof of Theorem 5.2, it is nothard to see that it gives the correct answer. Example 5.6.
Continuing Examples 5.3 and 5.5, the chosen ground state yields the columns431 31 21 2 and, after the cut, 1 31 2 41 2 3 . This tells us that the letter 1 appears in b , b , b , the letter 2 appears in b , b , b , and theletter 3 appear in b and b . This agrees with Figure 1.6. Energy and charge in type C The purpose of this section is to provide the proof of Theorem 1.1 for type C .6.1. The energy for type A fillings. We start with a brief discussion of the combinatorial R -matrix B ⊗ B → B ⊗ B for a tensor product of two type C columns. By the type C Pierirule [49], the decomposition of B ⊗ B into classical (non-affine) components is multiplicityfree. On the other hand, it was proved in [23] that the type C jeu de taquin due to Sheats[44] is compatible with the classical crystal operators. We conclude that the mentioned jeude taquin, when applied to our two-column situation, realizes the corresponding combinatorial R -matrix. Note that, in all situations considered below, the type C jeu de taquin is governedby the simpler rules of the classical one, in type A (see, e.g., [7]). Lemma 6.1.
Let b = b ⊗ b be a tensor product of two columns with all entries in [ n ] , wherethe height of b is at most that of b . Then the type A (1) n − and C (1) n (local) energies of b coincide.Proof. Consider the crystal B h , ⊗ B h , of type C (1) n , where n ≥ h ≥ h ≥
1. By the type C Pieri rule, the classical components of this crystal containing fillings with all entries in [ n ]have highest weights µ i = ( h + i, h − i ) (cid:48) , for i = 0 , . . . , min( h , n − h ). We will calculate the RYSTAL ENERGY VIA CHARGE 19 (local) energy on these components based on its definition (2.2), by setting the energy to 0 onthe component of highest weight µ (the usual normalization).For i = 1 , . . . , min( h , n − h ), consider b i ⊗ b i in B h , ⊗ B h , , where b i = n − h + 1... n , b i = 1 n − h − i + 2... n − h n − h + 1... n − i . The image b (cid:48) i ⊗ b (cid:48) i of b i ⊗ b i under the combinatorial R -matrix, constructed with jeu detaquin, is given by b (cid:48) i = n − h + 1... n , b (cid:48) i = 1 n − h − i + 2... n − i . Clearly, e acts RR on b i ⊗ b i and b (cid:48) i ⊗ b (cid:48) i , by changing 1 to 1. By type C insertion [23],which in this case is essentially just type A insertion (see, e.g., [7]), it is easy to see that b i ⊗ b i and e ( b i ⊗ b i ) lie in the components of highest weights µ i and µ i − , respectively. Therefore,the energy on the component of highest weight µ i is − i . This coincides with the type A energy(calculated via a similar procedure or via the type A charge). (cid:3) Proposition 6.2. If b is a tensor product of columns with all entries in [ n ] , then the type A n − and C n energies of b coincide. Furthermore, if the columns have weakly decreasing heights, theyequal the (type A n − or C n ) charge of b .Proof. When all entries are in [ n ], the jeu de taquin algorithms in types A n − and C n (realizingthe corresponding combinatorial R -matrices) work identically. Since the corresponding localenergies coincide by Lemma 6.1, the global energies coincide as well. But the type A n − energyof b is computed by the type A n − charge when the heights of the columns are weakly decreasingby Theorem 1.1 which was proven in Section 4, which clearly coincides with the correspondingtype C n charge. (cid:3) The conclusion of the proof.
We start by studying the behavior of the type C chargewith respect to the crystal operators and state a result in [31]. Proposition 6.3. [31]
The type C n charge is preserved by the crystal operators f , . . . , f n . Proposition 6.4.
Let B = B r N , ⊗ · · · ⊗ B r , be of type C n with r N ≥ r N − ≥ · · · ≥ r and b ∈ B . If ϕ ( b ) ≥ and ε ( b ) ≥ , then the type C n charge satisfies charge( e ( b )) = charge( b ) − . Proof.
Let b = b . . . b µ , and note that none of these columns contain both 1 and 1. Assumethat e changes the entry 1 in b j to 1. The condition ϕ ( b ) ≥ j >
1. Note that thecolumn b j − , and thus b Rj − , cannot contain 1 (otherwise, the entry 1 in b j would not be the leftmost unpaired 1). Similarly, the column b j +1 , and thus b Lj +1 , if they exist, cannot contain 1(otherwise, the entry 1 in b j would be paired with the mentioned 1).Let c := circ-ord( b ) with c = c L c R . . . c Lµ c Rµ , and d := circ-ord( e ( b )) with d = d L d R . . . d Lµ d Rµ (recall that the map circ-ord is defined by a slight modification of Algorithm 3.2). Assume that c Lj ( i ) = 1, which implies c Rj ( i ) = 1 (as we have no descents in the pair c Lj c Rj ; see Section 3.2). Weclaim that d Ll ( k ) = c Ll ( k ) and d Rl ( k ) = c Rl ( k ) for all k, l with the exception of d Lj ( i ) = d Rj ( i ) = 1.Indeed, all we need to check are the following, in this order: (1) the entries d Lj ( k ) and d Rj ( k )for k = 1 , . . . , i − d Lj ( i ) follows from the fact that b Rj − contains no 1; (3)the value of d Lj +1 ( i ), if this entry exists, follows from the fact that b Lj +1 contains no 1.By the above facts, we have the same descents in the fillings c and d , with the exceptionof the descent c Rj − ( i ) > c Lj ( i ) = 1, which corresponds to d Rj ( i ) = 1 > d Lj +1 ( i ), assuming that d Lj +1 ( i ) exists. The difference between the arm lengths of the mentioned descents is 2, whichconcludes the proof by (3.5). (cid:3) Proof of Theorem in type C . The proof is immediate based on Lemma 2.8, Theorem 5.2,and Propositions 6.2, 6.3, 6.4. (cid:3) Open problems
In this section we discuss several directions of research stemming from the results in thispaper. We intend to pursue some of them in the future.7.1.
Columns of types B and D . We believe that, in types B n and D n , the statistic in theRam–Yip formula for Macdonald polynomials at t = 0 can also be translated into a chargestatistic on the corresponding tensor product of KR crystals B k, of types B (1) n and D (1) n (rep-resented with KN columns), thus extending the results in [30]. We conjecture that the type B n and D n charge also agrees with the corresponding energy function, which would extendthe results in this paper. It should be possible to use a proof technique similar to the onein this paper, based on Lemma 2.8 from [42], to prove the conjecture; the proof would in-clude the generalization of the results in [31] to types B n and D n . These claims are supportedby [42, Corollary 9.5] expressing a type D n Macdonald polynomial at t = 0 in terms of thecorresponding energy function.From one point of view, the case of types B n and D n is easier than the one of type C n ,because the corresponding level 1 KR crystals are perfect. However, the construction of chargein types B n and D n displays additional complexity, due to some new aspects, that we nowdescribe. We start by referring to type B n , as type D n has all the complexity of type B n plusan additional one. For the corresponding KN columns, indexing the vertices of the crystals B ( ω i ) corresponding to the fundamental representations V ( ω i ), we refer to [17].The first new aspect in type B n is related to the splitting of the KR crystal B k, uponremoving the 0-arrows as the following direct sum of (classical) crystals: B ( ω k ) ⊕ B ( ω k − ) ⊕ · · · .This phenomenon manifests itself in the existence of descents between the left and the rightcolumns of a split KN column, which was not the case in type C n . We illustrate this based onthe following example. RYSTAL ENERGY VIA CHARGE 21
Example 7.1.
Let µ = ω in B . The KR crystal B , contains a vertex indexed by the KNcolumn b = 33 ∈ B ( ω ) ⊂ B , (cid:39) B ( ω ) ⊕ B ( ω ) ⊕ B ( ω ) . Instead of constructing the split column, we first construct from b the following “extended” KNcolumn of height 4, in order to make the columns corresponding to the components of B , ofthe same height: (cid:98) b = 1331 ;this construction is based on a procedure in [41, Section 3.4]. Only at this point we constructthe split column. We claim that the doubling procedure in Definition 2.2 can be extended,but we still match the entries in I = { > } with certain entries “preceding” them; the onlydifference is that this now requires us to go counterclockwise around the circle. So we obtain J = { , } , which gives the following splitting of (cid:98) b by the usual rule: (cid:98) b L (cid:98) b R = 2 14 33 41 2 . Finally, we construct c = c L c R = circ-ord( (cid:98) b L (cid:98) b R ) by the usual rule (see Section 3.2), where c L = (cid:98) b L : c = 2 34 23 11 4 . The energy of b in B , is −
1. This agrees with the charge of (cid:98) b L (cid:98) b R , computed via the sum in (3.5);indeed, there are two descents in c , whose arm lengths are 1, so charge( (cid:98) b L (cid:98) b R ) = (1 + 1) = 1.The second new aspect in type B n is the fact that not always a descent in the filling obtainedvia the procedure “circ-ord” contributes half its arm length to the charge. We illustrate thiswith another example. Example 7.2.
Let µ = 2 ω in type B , so B µ = B , ⊗ B , . Consider the pair of KN columns b b = 1 12 2in B µ . The corresponding fillings with split columns b = b L b R b L b R and c = circ-ord( b ) = c L c R c L c R coincide, and they are 1 1 1 12 2 2 2 . The corresponding energy is −
2, but in c we have only one descent of arm length 2, so wecannot use the sum in (3.5) to compute the charge. This problem occurs because, for all theentries i between 2 and 2 in circular order (namely i ∈ { , } ), either i or ı are above 2 in c R .This forces the descent 2 > The above example suggests that in type B n we need to modify the definition of charge bythe sum in (3.5) as follows. We use the same notation c = c L c R · · · c Lµ c Rµ as above. Let Des (cid:48) ( c )denote the descents of the form m = c Rj ( i ) > c Lj +1 ( i ) = m such that, for any k = 1 , . . . , m − k or k in c Rj [1 , i − B n charge is given by the following formula:(7.1) 12 (cid:88) γ ∈ Des( c ) \ Des (cid:48) ( c ) arm( γ ) + (cid:88) γ ∈ Des (cid:48) ( c ) arm( γ ) . We said above that in type D n we have additional complexity still. We explain this usinganother example. Example 7.3.
Let µ = 2 ω in type D . The filling with split columns c = b = b L b R b L b R =3 3 3 3 in B µ has no descents, but the corresponding energy is −
1. The reason for this isthat the values 3 and 3 are incomparable in type D , so the pair 3 3 in c needs to contribute 1to the charge. In fact, the definition of charge needs to be adjusted even more, as the followingmore subtle example shows.Let µ = 2 ω in D . Consider the following filling with split columns in B µ : c = b = b L b R b L b R = 3 3 4 44 4 3 3 . This filling has only ascents or equal entries next to each other in a row. However, the corre-sponding energy is −
1, so the charge needs to be 1. The reason for which the charge is not0, meaning that b is not a split KN tableau, is that the two middle columns in b representa forbidden configuration in type D . (Recall from [17] that, in type D , in addition to therow monotonicity condition for the split tableau in types B and C , there are extra conditionsfor a sequence of KN columns to form a KN tableau, and these are given in terms of certainforbidden configurations for a pair formed by a right column and the next left column.)7.2. Rows of types B , D , and C . In type A , Nakayashiki and Yamada [37] showed that theLascoux–Sch¨utzenberger charge statistic is related to the energy function on tensor products ofboth rows and columns. In particular, in [37, Proposition 3.23] they provide an explicit relationbetween the energy function on rows and columns by giving a bijection of each set to the setof semistandard Young tableaux which preserves the statistics.We expect a similar result to hold in the other classical types (with slight modifications).This claim is motivated by [28, Theorem 10.10], which relates the one-dimensional configurationsum in the large rank limit for columns to the one for rows in the dual type. More explicitly,denote by X Y n λ,B ( q ) the one-dimensional configuration sum, that is, the sum of q D ( b ) over allhighest weight elements b in the crystal B of weight λ and type Y n , graded by the energy D ( b ). Then up to an overall power of q , X B n λ,B ( q ) and X C n λ (cid:48) ,B (cid:48) ( q − ) are equal for large rank n .Here, if B = B r ,s ⊗ · · · ⊗ B r L ,s L then B (cid:48) = B s ,r ⊗ · · · ⊗ B s L ,r L and λ (cid:48) is the transpose of λ (where we identify dominant weights with partitions). Note that X B n λ,B ( q ) = X D n λ,B ( q ) when n islarge [28, 50].In particular, if the charge for columns of types B n or D n were known as outlined in Sec-tion 7.1 then, using B = B r , ⊗· · ·⊗ B r L , , the relation between X B n λ,B ( q ) and X C n λ (cid:48) ,B (cid:48) ( q − ) wouldprovide a way to obtain the charge in type C n also for single rows (as opposed to single columnsas treated in this paper). Computer experiments using Sage [47] indicate that this duality notonly holds on the level of configuration sums but also on the level of individual terms. We
RYSTAL ENERGY VIA CHARGE 23 expect that the highest weight crystal elements for both columns and rows are in bijection withcertain oscillating tableaux, as introduced in [1], due to the fact that the recording tableaux forthe Robinson–Schensted algorithm for types B n , C n , D n as introduced in [23, 27] are given bythese tableaux. If this conjecture were to be true, then we would be able to obtain the chargefor single rows from the charge for single columns via this bijection in the stable range (that is,large n ).7.3. Arbitrary order of columns and arbitrary types.
Consider a composition r =( r N , . . . , r ), and the crystal B := B r N , ⊗ · · · ⊗ B r , in types A and C . Since the chargeis defined only if r is a partition, the computation via the charge of the energy function onsome vertex b in B must proceed in general indirectly, by first appropriately commuting thecolumns of b via the combinatorial R -matrix (which preserves the energy). However, it is de-sirable to compute the energy directly on B . In addition, we would like to generalize to B ofarbitrary type, i.e., to find a statistic on B expressing the energy, which can be computed usingonly the combinatorial data corresponding to a vertex, instead of various paths in B .Let µ ( r ) be the partition obtained by reordering the parts of r . The Ram–Yip formula forthe Macdonald polynomial P µ ( r ) ( x ; q,
0) can be written in a way that is compatible with thecomposition r rather than the partition µ ( r ). Recall from [30, Proposition 2.7] that the termsin the above formula (for arbitrary type) correspond to chains of Weyl group elements givingrise to paths in the quantum Bruhat graph [2]; these chains are partitioned into segmentscorresponding to the parts of r , and the down steps (in Bruhat order) are measured by astatistic called level . In [31] we defined crystal operators on the above chains, both classicalones f i , i >
0, and the affine one f . We called this construction the quantum alcove model , as itgeneralizes the alcove model in [32, 33]. The main conjecture is that the new model uniformlydescribes tensor products of column shape KR crystals, for all untwisted affine types. Theconjecture is proved in the same paper in types A and C , by showing that the bijections in[30] from the objects of the quantum alcove model to tensor products of the corresponding KNcolumns are affine crystal isomorphisms. With S. Naito and M. Shimozono, we are working ona uniform proof of the conjecture for all untwisted affine types. This will imply that the levelstatistic in the Ram-Yip formula expresses the corresponding energy function. In particular,we would obtain a generalization of the combinatorial formula (1.1) to arbitrary type, wherethe charge is replaced by the level.7.4. Demazure crystals and Macdonald polynomials.
Ion [12] showed that, in simply-laced types, one can identify a Macdonald polynomial at t = 0, namely P µ ( x ; q, P µ ( x ; q,
0) oftype C , together with the decomposition of the corresponding tensor product of KR crystalswith only the Demazure arrows (studied in Section 5) suggest that, in general, P µ ( x ; q,
0) couldbe identified with a sum of Demazure characters. If so, one would want to determine themexplicitly and, in particular, to know if all correspond to shifts in the affine Weyl group.As Ion’s result and the Ram–Yip formula were given for non-symmetric Macdonald poly-nomials as well, we can also ask about the generalization of Ion’s result to arbitrary types inthe non-symmetric case. There are indications that the combinatorial formula (1.1), which wasconjectured to have a version for arbitrary type (see Section 7.3), should be replaced with onewhose right-hand side is a sum over a subset of the corresponding tensor product of KR crystals.If so, it would be desirable to describe this subset in an explicit way. See also [42, Section 9.2].
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Department of Mathematics and Statistics, State University of New York at Albany, Albany,NY 12222, U.S.A.
E-mail address : [email protected] URL : Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616-8633, U.S.A.
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