Generalized Energy Statistics and Kostka--Macdonald Polynomials
aa r X i v : . [ m a t h . QA ] A p r Generalized Energy Statistics andKostka–Macdonald Polynomials a Anatol N. Kirillov and b Reiho Sakamoto a Research Institute for Mathematical Sciences,Kyoto University, Sakyo-ku, Kyoto, 606-8502, [email protected] b Department of Physics, Tokyo University of Science,Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, [email protected]
Abstract:
We give an interpretation of the t = 1 specialization of themodified Macdonald polynomial as a generating function of the energystatistics defined on the set of paths arising in the context of Box-BallSystems (BBS-paths for short). We also introduce one parameter gen-eralizations of the energy statistics on the set of BBS-paths which all,conjecturally, have the same distribution. R´esum´e:
Nous donnons une int´erpr´etation de la sp´ecialisation `a t = 1du polynˆome de Macdonald modifi´e comme fonction g´en´eratrice desstatistiques d’´energie d´efinies sur l’ensemble des chemins qui apparais-sent dans la th´eorie des Syst`emes BBS (BBS-chemins). Nous pr´esentons´egalement des g´en´eralisations `a un param`etre de la statistique d’´energiesur les chemins BBS qui toutes, conjecturalement, ont la mˆeme distribu-tion. Key words: modified Macdonald polynomials, box-ball systems.1
Introduction
The purpose of the present paper is two-fold. First of all we would like to draw atten-tion to a rich combinatorics hidden behind the dynamics of Box-Ball Systems, andsecondly, to connect the former with the theory of modified Macdonald polynomials.More specifically, our final goal is to give an interpretation of the Kostka–Macdonaldpolynomials K λ,µ ( q, t ) as a refined partition function of a certain box-ball systemsdepending on initial data λ and µ .Box-Ball Systems (BBS for short) were invented by Takahashi–Satsuma [29, 28]as a wide class of discrete integrable soliton systems. In the simplest case, BBSare described by simple combinatorial procedures using boxes and balls. One cansee the simplest but still very interesting examples of the BBS by the free softwareavailable at [26]. Despite its simple outlook, it is known that the BBS have variousremarkably deep properties: • Local time evolution rule of the BBS coincides with the isomorphism of thecrystal bases [7, 2]. Thus the BBS possesses quantum integrability. • BBS are ultradiscrete (or tropical) limit of the usual soliton systems [30, 20].Thus the BBS possesses classical integrability at the same time. • Inverse scattering formalism of the BBS [19] coincides with the rigged con-figuration bijection originating in completeness problem of the Bethe states[14, 16], see also [25].Let us say a few words about the main results of this note. • We will identify the space of states of a BBS with the corresponding weightsubspace in the tensor product of fundamental (or rectangular) representationsof the Lie algebra gl ( n ) . • In the case of statistics tau , our main result can be formulated as a computationof the corresponding partition function for the BBS in terms of the values ofthe Kostka–Macdonald polynomials at t = 1 . • In the case of the statistics energy , our result can be formulated as an inter-pretation of the corresponding partition function for the BBS as the q -weightmultiplicity of a certain irreducible representation of the Lie algebra gl ( n ) inthe tensor product of the fundamental representations. We expect that thesame statement is valid for the BBS corresponding to the tensor product ofrectangular representations.Let us remind that a q -analogue of the multiplicity of a highest weight λ inthe tensor product N La =1 V s a ω ra of the highest weight s a ω r a , a = 1 , . . . , L, irreducible representations V s a ω ra of the Lie algebra gl ( n ) is defined as q -Mult [ V λ : L O a =1 V s a ω ra ] = X η K η,R K η,λ ( q ) , K η,R stands for the parabolic Kostka number corresponding to the se-quence of rectangles R := { ( s r a a ) } a =1 ,...,L , see e.g. [15], [18].A combinatorial description of the modified Macdonald polynomials has beenobtained by Haglund–Haiman–Loehr [5]. In Section 5 we give an interpretation oftwo Haglund’s statistics in the context of the box-ball systems, i.e., in terms of theBBS-paths. Namely, we identify the set of BBS paths of weight α with the set P ( α )which is the weight α component in the tensor product of crystals corresponding tovector representations. We have observed that from the proof given in [5] one canprove the following identity X p ∈P ( α ) q inv µ ( p ) t maj µ ( p ) = X η ⊢| µ | K η,α ˜ K η,µ ( q, t ) , (1)see Proposition 6.2 and Corollary 6.3. One of the main problems we are interestedin is to generalize the identity Eq.(1) on more wider set of the BBS-paths.Our result about connections of the energy partition functions for BBS and q -weight multiplicities suggests a deep hidden connections between partition functionsfor the BBS and characters of the Demazure modules, solutions to the q -differenceToda equations, cf.[3], ... .As an interesting open problem we want to give raise a question about an in-terpretation of the sums P η K η,R K η,λ ( q, t ) , where K η,λ ( q, t ) denotes the Kostka–Macdonald polynomials [21], as refined partition functions for the BBS correspond-ing to the tensor product of rectangular representations R = { ( s r a a ) } ≤ a ≤ n . In otherwords, one can ask: what is a meaning of the second statistics (see [5]) in theKashiwara theory [11] of crystal bases (of type A) ?This paper is abbreviated and updated version of our paper [17]. The mainnovelty of the present paper is the definition of a one parameter family of statisticson the set of BBS-paths which generalizes those introduced in [17], see Conjecture7.2. It conjecturally gives a new family of MacMahonian statistics on the set oftransportation matrices, see [15].Organization of the present paper is as follows. In Section 2 we outlook the basicdefinitions and facts related to the Kashiwara’s theory of crystal base in the case oftype A (1) n . We also remind definitions of the combinatorial R -matrix and definitionof the energy function. We illustrate definitions by simple example. In Section 3,we introduce the energy statistics and the set of the BBS. In Section 4 we reminddefinition of box-ball systems and state some of their simplest properties. In Section5 we remind definition of the Haglund’s statistics and give their interpretation interms of the BBS-paths. Sections 6 and 7 contain our main results and conjectures.In particular it is not difficult to see that Haglund’s statistics maj µ and inv µ do notcompatible with the Kostka–Macdonald polynomials for general partitions λ and µ .In Section 6 we state a conjecture which describes the all pairs of partitions ( λ, µ )for those the restriction of the Haglund–Haiman–Loehr formula on the set of highestweight paths of shape µ coincide with the Kostka–Macdonald polynomial ˜ K λ,µ ( q, t ).3 Kirillov–Reshetikhin crystal A (1) n type crystal Let W ( r ) s be a U ′ q ( g ) Kirillov–Reshetikhin module, where we shall consider the case g = A (1) n . The module W ( r ) s is indexed by a Dynkin node r ∈ I = { , , . . . , n } and s ∈ Z > . As a U q ( A n )-module, W ( r ) s is isomorphic to the irreducible modulecorresponding to the partition ( s r ). For arbitrary r and s , the module W ( r ) s is knownto have crystal bases [11, 10], which we denote by B r,s . As the set, B r,s is consistingof all column strict semi-standard Young tableaux of depth r and width s over thealphabet { , , . . . , n + 1 } .For the algebra A n , let P be the weight lattice, { Λ i ∈ P | i ∈ I } be the funda-mental roots, { α i ∈ P | i ∈ I } be the simple roots, and { h i ∈ Hom Z ( P, Z ) | i ∈ I } bethe simple coroots. As a type A n crystal, B = B r,s is equipped with the Kashiwaraoperators ˜ e i , ˜ f i : B −→ B ∪ { } and wt : B −→ P ( i ∈ I ) satisfying˜ f i ( b ) = b ′ ⇐⇒ ˜ e i ( b ′ ) = b if b, b ′ ∈ B, wt (cid:0) ˜ f i ( b ) (cid:1) = wt( b ) − α i if ˜ f i ( b ) ∈ B, h h i , wt( b ) i = ϕ i ( b ) − ε i ( b ) . Here h· , ·i is the natural pairing and we set ε i ( b ) = max { m ≥ | ˜ e mi b = 0 } and ϕ i ( b ) = max { m ≥ | ˜ f mi b = 0 } . Actions of the Kashiwara operators ˜ e i , ˜ f i for i ∈ I coincide with the one described in [12]. Since we do not use explicit forms of theseoperators, we omit the details. See [23] for complements of this section. Note thatin our case A n , we have P = Z n +1 and α i = ǫ i − ǫ i +1 where ǫ i is the i -th canonicalunit vector of Z n +1 . We also remark that wt( b ) = ( λ , · · · , λ n +1 ) is the weight of b ,i.e., λ i counts the number of letters i contained in tableau b .For two crystals B and B ′ , one can define the tensor product B ⊗ B ′ = { b ⊗ b ′ | b ∈ B, b ′ ∈ B ′ } . The actions of the Kashiwara operators on tensor product havesimple form. Namely, the operators ˜ e i , ˜ f i act on B ⊗ B ′ by˜ e i ( b ⊗ b ′ ) = (cid:26) ˜ e i b ⊗ b ′ if ϕ i ( b ) ≥ ε i ( b ′ ) b ⊗ ˜ e i b ′ if ϕ i ( b ) < ε i ( b ′ ) , ˜ f i ( b ⊗ b ′ ) = (cid:26) ˜ f i b ⊗ b ′ if ϕ i ( b ) > ε i ( b ′ ) b ⊗ ˜ f i b ′ if ϕ i ( b ) ≤ ε i ( b ′ ) , and wt( b ⊗ b ′ ) = wt( b ) + wt( b ′ ). We assume that 0 ⊗ b ′ and b ⊗ R : B r,s ⊗ B r ′ ,s ′ ∼ → B r ′ ,s ′ ⊗ B r,s .We call this map (classical) combinatorial R and usually write the map R simplyby ≃ .Let us consider the affinization of the crystal B . As the set, it isAff( B ) = { b [ d ] | b ∈ B, d ∈ Z } . (2)There is also explicit algorithm for actions of the affine Kashiwara operators ˜ e ,˜ f in terms of the promotion operator [27]. For the tensor product b [ d ] ⊗ b ′ [ d ′ ] ∈ B ) ⊗ Aff( B ′ ), we can lift the (classical) combinatorial R to affine case as follows: b [ d ] ⊗ b ′ [ d ′ ] R ≃ ˜ b ′ [ d ′ − H ( b ⊗ b ′ )] ⊗ ˜ b [ d + H ( b ⊗ b ′ )] , (3)where b ⊗ b ′ ≃ ˜ b ′ ⊗ ˜ b is the isomorphism of (classical) combinatorial R . The function H ( b ⊗ b ′ ) is called the energy function and defined by a certain set of axioms. We willgive explicit forms of the combinatorial R and energy function in the next section. R and energy function We give an explicit description of the combinatorial R -matrix (combinatorial R forshort) and energy function on B r,s ⊗ B r ′ ,s ′ . To begin with we define few terminologiesabout Young tableaux. Denote rows of a Young tableaux Y by y , y , . . . y r from topto bottom. Then row word row ( Y ) is defined by concatenating rows as row ( Y ) = y r y r − . . . y . Let x = ( x , x , . . . ) and y = ( y , y , . . . ) be two partitions. We defineconcatenation of x and y by the partition ( x + y , x + y , . . . ). Proposition 2.1 ([27]) b ⊗ b ′ ∈ B r,s ⊗ B r ′ ,s ′ is mapped to ˜ b ′ ⊗ ˜ b ∈ B r ′ ,s ′ ⊗ B r,s under the combinatorial R , i.e., b ⊗ b ′ R ≃ ˜ b ′ ⊗ ˜ b, (4) if and only if ( b ′ ← row ( b )) = (˜ b ← row (˜ b ′ )) . (5) Moreover, the energy function H ( b ⊗ b ′ ) is given by the number of nodes of ( b ′ ← row ( b )) outside the concatenation of partitions ( s r ) and ( s ′ r ′ ) . For special cases of B ,s ⊗ B ,s ′ , the function H is called unwinding number in[22]. Explicit values for the case b ⊗ b ′ ∈ B , ⊗ B , are given by H ( b ⊗ b ′ ) = χ ( b < b ′ )where χ (True) = 1 and χ (False) = 0.In order to describe the algorithm for finding ˜ b and ˜ b ′ from the data ( b ′ ← row ( b )),we introduce a terminology. Let Y be a tableau, and Y ′ be a subset of Y such that Y ′ is also a tableau. Consider the set theoretic subtraction θ = Y \ Y ′ . If the numberof nodes contained in θ is r and if the number of nodes of θ contained in each rowis always 0 or 1, then θ is called vertical r -strip.Given a tableau Y = ( b ′ ← row ( b )), let Y ′ be the upper left part of Y whoseshape is ( s r ). We assign numbers from 1 to r ′ s ′ for each node contained in θ = Y \ Y ′ by the following procedure. Let θ be the vertical r ′ -strip of θ as upper as possible.For each node in θ , we assign numbers 1 through r ′ from the bottom to top. Nextwe consider θ \ θ , and find the vertical r ′ strip θ by the same way. Continue thisprocedure until all nodes of θ are assigned numbers up to r ′ s ′ . Then we apply inversebumping procedure according to the labeling of nodes in θ . Denote by u the integerwhich is ejected when we apply inverse bumping procedure starting from the nodewith label 1. Denote by Y the tableau such that ( Y ← u ) = Y . Next we apply5nverse bumping procedure starting from the node of Y labeled by 2, and obtainthe integer u and tableau Y . We do this procedure until we obtain u r ′ s ′ and Y r ′ s ′ .Finally, we have ˜ b ′ = ( ∅ ← u r ′ s ′ u r ′ s ′ − · · · u ) , ˜ b = Y r ′ s ′ . (6) Example 2.2
Consider the following tensor product: b ⊗ b ′ = 1 1 42 3 6 ⊗ ∈ B , ⊗ B , . From b , we have row ( b ) = 236114, hence we have ← = 1 1 3 4 . Here subscripts of each node indicate the order of inverse bumping procedure. Forexample, we start from the node 4 and obtain ← = 1 1 3 42 2 63 34 45 , therefore , Y = 1 2 3 4 , u = 1 . Next we start from the node 4 of Y . Continuing in this way, we obtain u u · · · u =321421 and Y = 3 3 44 5 6 . Since ( ∅ ← ⊗ ≃ ⊗ , H ⊗ = 3 . Note that the energy function is derived from the concatenation of shapes of b and b ′ , i.e., . 6 Energy statistics and its generalizations on theset of paths
For a path b ⊗ b ⊗ · · · ⊗ b L ∈ B r ,s ⊗ B r ,s ⊗ · · · ⊗ B r L ,s L , let us define elements b ( i ) j ∈ B r j ,s j for i < j by the following isomorphisms of the combinatorial R ; b ⊗ b ⊗ · · · ⊗ b i − ⊗ b i ⊗ · · · ⊗ b j − ⊗ b j ⊗ · · ·≃ b ⊗ b ⊗ · · · ⊗ b i − ⊗ b i ⊗ · · · ⊗ b ( j − j ⊗ b ′ j − ⊗ · · ·≃ · · ·≃ b ⊗ b ⊗ · · · ⊗ b i − ⊗ b ( i ) j ⊗ · · · ⊗ b ′ j − ⊗ b ′ j − ⊗ · · · , (7)where we have written b k ⊗ b ( k +1) j ≃ b ( k ) j ⊗ b ′ k assuming that b ( j ) j = b j .Define the statistics maj( p ) bymaj( p ) = X i For the path p ∈ B r ,s ⊗ B r ,s ⊗ · · · ⊗ B r L ,s L , define τ r,s by τ r,s ( p ) = maj( u ( r ) s ⊗ p ) , (10) where u ( r ) s is the highest element of B r,s . Here the highest element u ( r ) s ∈ B r,s is the tableau whose i -th row is occupied byintegers i . For example, u (3)4 = 1 1 1 12 2 2 23 3 3 3 . In particular, the statistics τ r, on B , type paths a ∈ ( B , ) ⊗ L has the following form; τ r, ( a ) = L · χ ( r < a ) + L − X i =1 ( L − i ) χ ( a i < a i +1 ) , (11)where a denotes the first letter of the path a . Note that τ , is a special case of thetau functions for the box-ball systems [20, 24] which originates as an ultradiscretelimit of the tau functions for the KP hierarchy [9]. Definition 3.2 For composition µ = ( µ , µ , · · · , µ n ) , write µ [ i ] = P ij =1 µ j withconvention µ [0] = 0 . Then we define a generalization of τ r, by τ r, µ ( a ) = n X i =1 τ r, ( a [ i ] ) , (12)7 here a [ i ] = a µ [ i − +1 ⊗ a µ [ i − +2 ⊗ · · · ⊗ a µ [ i ] ∈ ( B , ) ⊗ µ i . (13)Note that we have a = a [1] ⊗ a [2] ⊗ · · · ⊗ a [ n ] , i.e., the path a is partitioned accordingto µ . In this section, we summarize basic facts about the box-ball system in order toexplain physical origin of τ , . For our purpose, it is convenient to express theisomorphism of the combinatorial R : a ⊗ b ≃ b ′ ⊗ a ′ by the following vertex diagram: a b ′ b a ′ .Successive applications of the combinatorial R is depicted by concatenating thesevertices.Following [7, 2], we define time evolution of the box-ball system T ( a ) l . Let u ( a ) l, = u ( a ) l ∈ B a,l be the highest element and b i ∈ B r i ,s i . Define u ( a ) l,j and b ′ i ∈ B r i ,s i by thefollowing diagram. u ( a ) l, b b ′ u ( a ) l, b b ′ u ( a ) l, ·········· u ( a ) l,L − b L b ′ L u ( a ) l,L (14) u ( a ) l,j are usually called carrier and we set u ( a ) l, := u ( a ) l . Then we define operator T ( a ) l by T ( a ) l ( b ) = b ′ = b ′ ⊗ b ′ ⊗ · · · ⊗ b ′ L . (15)Recently [25], operators T ( a ) l have used to derive crystal theoretical meaning of therigged configuration bijection.It is known ([19] Theorem 2.7) that there exists some l ∈ Z > such that T ( a ) l = T ( a ) l +1 = T ( a ) l +2 = · · · (=: T ( a ) ∞ ) . (16)If the corresponding path is b ∈ ( B , ) ⊗ L , we have the following combinatorialdescription of the box-ball system [29, 28]. We regard 1 ∈ B , as an empty box ofcapacity 1, and i ∈ B , as a ball of label (or internal degree of freedom) i containedin the box. Then we have: Proposition 4.1 ([7]) For a path b ∈ ( B , ) ⊗ L of type A (1) n , T (1) ∞ ( b ) is given by thefollowing procedure.1. Move every ball only once. . Move the leftmost ball with label n + 1 to the nearest right empty box.3. Move the leftmost ball with label n + 1 among the rest to its nearest right emptybox.4. Repeat this procedure until all of the balls with label n + 1 are moved.5. Do the same procedure 2–4 for the balls with label n .6. Repeat this procedure successively until all of the balls with label are moved. There are extensions of this box and ball algorithm corresponding to generalizationsof the box-ball systems with respect to each affine Lie algebra, see e.g., [8]. Using thisbox and ball interpretation, our statistics τ , ( b ) admits the following interpretation. Theorem 4.2 ([20] Theorem 7.4) For a path b ∈ ( B , ) ⊗ L of type A (1) n , τ , ( b ) coincides with number of all balls , · · · , n + 1 contained in paths b , T (1) ∞ ( b ) , · · · , ( T (1) ∞ ) L − ( b ) . Example 4.3 Consider the path p = a ⊗ b where a = 4311211111, b = 4321111111.Note that we omit all frames of tableaux of B , and symbols for tensor product. Wecompute τ (10 , ( p ) by using Theorem 4.2. According to Proposition 4.1, the timeevolutions of the paths a and b are as follows:4 3 1 1 2 1 1 1 1 11 1 4 3 1 2 1 1 1 11 1 1 1 4 1 3 2 1 11 1 1 1 1 4 1 1 3 21 1 1 1 1 1 4 1 1 11 1 1 1 1 1 1 4 1 11 1 1 1 1 1 1 1 4 11 1 1 1 1 1 1 1 1 4 4 3 2 1 1 1 1 1 1 11 1 1 4 3 2 1 1 1 11 1 1 1 1 1 4 3 2 11 1 1 1 1 1 1 1 1 41 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1Here the left and right tables correspond to a and b , respectively. Rows of left (resp.right) table represent a , T (1) ∞ ( a ), · · · , ( T (1) ∞ ) L ( a ) (resp., those for b ) from top tobottom. Counting letters 2, 3 and 4 in each table, we have τ , ( a ) = 16, τ , ( b ) = 10and we get τ , , ( p ) = 16 + 10 = 26, which coincides with the computation byEq.(11). Meanings of the above two dynamics corresponding to paths a and b aresummarized as follows:( a ) Dynamics of the path a . In the first two rows, there are two solitons (lengthtwo soliton 43 and length one soliton 2), and in the lower rows, there are alsotwo solitons (length one soliton 4 and length two soliton 32). This is scatteringof two solitons. After the scattering, soliton 4 propagates at velocity one andsoliton 32 propagates at velocity two without scattering.( b ) Dynamics of the path b . This shows free propagation of one soliton of lengththree 432 at velocity three. 9 Haglund’s statistics Tableaux language description For a given path a = a ⊗ a ⊗ · · · ⊗ a L ∈ ( B , ) ⊗ L , associate tabloid t of shape µ whose reading word coincides with a . Forexample, to path p = abcdefgh and the composition µ = (3 , , 3) one associates thetabloid c b ae dh g f . (17)Denote the cell at the i -th row, j -th column (we denote the coordinate by ( i, j ))of the tabloid t by t ij . Attacking region of the cell at ( i, j ) is all cells ( i, k ) with k < j or ( i + 1 , k ) with k > j . In the following diagram, gray zonal regions are theattacking regions of the cell ( i, j ). ✚✚✚❂ ( i, j )Follow [5], define | Inv ij | by | Inv ij | = { ( k, l ) ∈ attacking region for ( i, j ) | t kl > t ij } . (18)Then we define | Inv µ ( a ) | = X ( i,j ) ∈ µ | Inv ij | . (19)If we have t ( i − j < t ij , then the cell ( i, j ) is called by descent . Then defineDes µ ( a ) = X all descent ( i,j ) ( µ i − j ) . (20)Note that ( µ i − j ) is the arm length of the cell ( i, j ). Path language description Consider two paths a (1) , a (2) ∈ ( B , ) ⊗ µ . We denoteby a (1) ⊗ a (2) = a ⊗ a ⊗ · · · ⊗ a µ . Then we defineInv ( µ,µ ) ( a (1) , a (2) ) = µ X k =1 k + µ − X i = k +1 χ ( a k < a i ) . (21)For more general cases a (1) ∈ ( B , ) ⊗ µ and a (2) ∈ ( B , ) ⊗ µ satisfying µ > µ , wedefine Inv ( µ ,µ ) ( a (1) , a (2) ) := Inv ( µ ,µ ) ( a (1) , ⊗ ( µ − µ ) ⊗ a (2) ) . (22)10hen the above definition of | Inv µ ( a ) | is equivalent to | Inv µ ( a ) | = n − X i =1 Inv ( µ i ,µ i +1 ) . (23)Consider two paths a (1) ∈ ( B , ) ⊗ µ and a (2) ∈ ( B , ) ⊗ µ satisfying µ ≥ µ .Denote a = a (1) ⊗ a (2) . Then defineDes ( µ ,µ ) ( a ) = µ X k = µ − µ +1 ( k − ( µ − µ ) − χ ( a k < a k + µ ) . (24)For the tableau T of shape µ corresponding to the path a , we defineDes µ ( T ) = n X i =1 Des ( µ i ,µ i +1 ) ( a [ i ] ⊗ a [ i +1] ) . (25) Definition 5.1 ([4]) For a path a , statistics maj µ is defined by maj µ ( a ) = µ X i =1 maj( t ,i ⊗ t ,i ⊗ · · · ⊗ t µ ′ i ,i ) . (26) and inv µ ( a ) is defined by inv µ ( a ) = | Inv µ ( a ) | − Des µ ( a ) . (27)If we associate to a given path p ∈ P ( λ ) with the shape µ tabloid T , we sometimeswrite maj µ ( p ) = maj( T ) and inv µ ( p ) = inv( T ). Let ˜ H µ ( x ; q, t ) be the (integral form) modified Macdonald polynomials where x stands for infinitely many variables x , x , · · · . Here ˜ H µ ( x ; q, t ) is obtained by simpleplethystic substitution (see, e.g., section 2 of [6]) from the original definition of theMacdonald polynomials [21]. Schur function expansion of ˜ H µ ( x ; q, t ) is given by˜ H µ ( x ; q, t ) = X λ ˜ K λ,µ ( q, t ) s λ ( x ) , (28)where ˜ K λ,µ ( q, t ) stands for the following transformation of the Kostka–Macdonaldpolynomials: ˜ K λ,µ ( q, t ) = t n ( µ ) K λ,µ ( q, t − ) . (29)Here we have used notation n ( µ ) = P i ( i − µ i . Then the celebrated Haglund–Haiman–Loehr (HHL) formula is as follows.11 heorem 6.1 ([5]) Let σ : µ → Z > be the filling of the Young diagram µ bypositive integers Z > , and define x σ = Q u ∈ µ x σ ( u ) . Then the Macdonald polynomial ˜ H µ ( x ; q, t ) have the following explicit formula: ˜ H µ ( x ; q, t ) = X σ : µ → Z > q inv( σ ) t maj( σ ) x σ . (30)From the HHL formula, we can show the following formula. Proposition 6.2 For any partition µ and composition α of the same size, one has X p ∈P ( α ) q inv µ ( p ) t maj µ ( p ) = X η ⊢| µ | K η,α ˜ K η,µ ( q, t ) , (31) where P ( α ) stands for the set of type B , paths of weight α = ( α , α , . . . , α n +1 ) and η runs over all partitions of size | µ | . Corollary 6.3 The (modified) Macdonald polynomial ˜ H µ ( x ; q, t ) have the followingexpansion in terms of the monomial symmetric functions m λ ( x ) : ˜ H µ ( x ; q, t ) = X λ ⊢| µ | X p ∈P ( λ ) q inv µ ( p ) t maj µ ( p ) m λ ( x ) , (32) where λ runs over all partitions of size | µ | . To find combinatorial interpretation of the Kostka–Macdonald polynomials ˜ K λ,µ ( q, t )remains significant open problem. Among many important partial results aboutthis problem, we would like to mention the following theorem also due to Haglund–Haiman–Loehr: Theorem 6.4 ([5] Proposition 9.2) If µ ≤ , we have ˜ K λ,µ ( q, t ) = X p ∈P + ( λ ) q inv µ ( p ) t maj µ ( p ) , (33) where P + ( λ ) is the set of all highest weight elements of P ( λ ) according to the readingorder explained in Eq.(17). It is interesting to compare this formula with the formula obtained by S. Fishel [1],see also [14], [18].Concerning validity of the formula Eq.(33), we state the following conjecture. Conjecture 6.5 Explicit formula for the Kostka–Macdonald polynomials ˜ K λ,µ ( q, t ) = X p ∈P + ( λ ) q inv µ ( p ) t maj µ ( p ) . (34) is valid if and only if at least one of the following two conditions is satisfied.(i) µ ≤ and µ ≤ .(ii) λ is a hook shape. Generating function of tau functions In [17], we give an elementary proof for special case t = 1 of the formula Eq.(31) inthe following form. Theorem 7.1 Let α be a composition and µ be a partition of the same size. Then, X p ∈P ( α ) q maj µ ′ ( p ) = X η ⊢| µ | K η,α K η,µ ( q, . (35) Conjecture 7.2 Let α be a composition and µ be a partition of the same size. Then, q − P i>r α i X p ∈P ( α ) q τ r, µ ( p ) = X η ⊢| µ | K η,α ˜ K η,µ ( q, . (36)This conjecture contains Conjecture 5.8 of [17] and Theorem 7.1 above as specialcases r = 1 and r = ∞ , respectively. Also, extensions for paths of more generalrepresentations without partition µ are discussed in Section 5.3 of [17]. Example 7.3 Let us consider case α = (4 , , 1) and µ = (4 , p and the corresponding value of tau function τ , , ( p ). For example, thetop left corner 111123 1 means p = 1 ⊗ ⊗ ⊗ ⊗ ⊗ τ , , ( p ) = 1.111123 1 111132 2 111213 2 111231 3 111312 2 111321 1112113 3 112131 4 112311 3 113112 3 113121 2 113211 2121113 4 121131 5 121311 4 123111 5 131112 4 131121 3131211 4 132111 3 211113 1 211131 2 211311 1 213111 2231111 3 311112 5 311121 4 311211 5 312111 6 321111 4Summing up, LHS of Eq.(36) is q − X p ∈P ((4 , , q τ , , ( p ) = q + 4 q + 7 q + 7 q + 7 q + 4which coincides with the RHS of Eq.(36). Compare this with τ , , data for the sameset of paths at Example 5.9 of [17]. Acknowledgements: The work of RS is supported by Grant-in-Aid for ScientificResearch (No.21740114), JSPS. References [1] S. Fishel, Statistics for special q, t -Kostka polynomials, Proc. Amer. Math. Soc. (1995) 2961–2969.[2] K. Fukuda, M. Okado and Y. Yamada, Energy functions in box-ball systems,Int. J. Mod. Phys. 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