On the expansion of certain vector-valued characters of U q ( gl n ) with respect to the Gelfand-Tsetlin basis
aa r X i v : . [ m a t h . R T ] S e p ON THE EXPANSION OF CERTAIN VECTOR-VALUED CHARACTERSOF U q ( gl n ) WITH RESPECT TO THE GELFAND-TSETLIN BASIS
VIDYA VENKATESWARAN
Abstract.
Macdonald polynomials are an important class of symmetric functions, withconnections to many different fields. Etingof and Kirillov showed an intimate connec-tion between these functions and representation theory: they proved that Macdonaldpolynomials arise as (suitably normalized) vector-valued characters of irreducible rep-resentations of quantum groups. In this paper, we provide a branching rule for thesecharacters. The coefficients are expressed in terms of skew Macdonald polynomials withplethystic substitutions. We use our branching rule to give an expansion of the charac-ters with respect to the Gelfand-Tsetlin basis. Finally, we study in detail the q = 0 case,where the coefficients factor nicely, and have an interpretation in terms of certain p -adiccounts. Introduction
Macdonald polynomials were originally discovered in the 1980s [10, 9], and have founda variety of uses in mathematics, appearing in a number of disparate fields (mathematicalphysics, combinatorics, representation theory and number theory, among others). Thesepolynomials have the key property of being invariant under all permutations of their n vari-ables. They are indexed by partitions λ with length at most n , and form an orthogonal basisfor the ring of symmetric polynomials with coefficients in C ( q, t ) with respect to a certaindensity function. The existence of such polynomials was proved by exhibiting particulardifference operators which have these polynomials as their eigenfunctions. Macdonald poly-nomials contain many important families as particular degenerations of the parameters q and t . In particular, the ubiquitous Schur functions are obtained by setting q = t ; crucially,these are characters of irreducible representations of GL n . Hall-Littlewood polynomials arerecovered in the limit q = 0, and these have interpretations as zonal spherical functions on p -adic groups. Some other important subfamilies are the monomial, elementary, and powersum symmetric functions.Given the various connections to representation theory, one might ask whether Macdonaldpolynomials arise as characters of certain irreducible representations. Etingof and Kirillovdiscovered such a realization in [2], where they demonstrate that Macdonald polynomialsare ratios of vector-valued characters of representations of the quantum group U q ( gl n ).Recall that the finite-dimensional, irreducible representations V λ of U g ( gl n ) are indexed by λ ∈ P ( n )+ = { ( λ , . . . , λ n ) : λ i − λ i +1 ∈ Z + } . Note that elements of P ( n )+ can be written as( a, a, . . . , a ) + e λ , where a ∈ C and e λ is a standard partition of length n . Let k ∈ N be fixed,then they show the existence of an intertwining operator (unique up to scaling):(1) φ ( k ) λ : V λ +( k − ρ → V λ +( k − ρ ⊗ U, where U ≃ V ( k − · ( n − , − ,..., − and ρ = (cid:0) n − , n − , . . . , − n (cid:1) . Note that U has the specialproperty that all weight subspaces are one-dimensional. Fix the normalization of φ ( k ) λ sothat v λ +( k − ρ → v λ +( k − ρ ⊗ u + · · · , where v λ +( k − ρ is a fixed non-zero highest weightvector for V λ +( k − ρ and u is a fixed non-zero vector in the (one-dimensional) weight zerosubspace of U . Consider the corresponding trace function of this operator: Research supported by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS-1204900. Φ ( k ) λ ( x , . . . , x n ) = Tr( φ ( k ) λ · x h ) ∈ C ( q )[ x , . . . , x n ] , where q h = q ( h + ··· + h n ) ∈ U q ( gl n ). Then Etingof and Kirillov proved the following intimateconnection between these trace functions and Macdonald polynomials: Theorem 1.1. [2]
The Macdonald polynomial P λ ( x ; q , q k ) is given by the ratio P λ ( x ; q , q k ) = Φ ( k ) λ ( x )Φ ( k )0 ( x ) . Moreover, there are formal power series e Φ λ ( x ; q, t ) ∈ C ( q, t )[[ x , . . . , x n ]] such that P λ ( x ; q , t ) = e Φ λ ( x ; q, t ) e Φ ( x ; q, t ) , and Φ ( k ) λ ( x , . . . , x n ) = e Φ λ ( x , . . . , x n ; q, q k ) . The two-parameter family e Φ λ ( x ; q, t ) can be realized as the trace function of an inter-twining operator analogous to (1), where V λ +( k − ρ and U are replaced by suitable infinite-dimensional, irreducible representations of C ( t ) ⊗ U q ( gl n ) (see [2] or Section 5 for details).In this paper we consider the expansion of the trace function Φ ( k ) ( x ) with respect to theGelfand-Tsetlin basis of V λ +( k − ρ . We give explicit combinatorial formulas for the diagonalcoefficients of the intertwining operator φ ( k ) with respect to this basis, as sums of products ofwell-known rational functions appearing in symmetric function theory, specialized to t = q k .Inspired by Kashiwara’s theory of crystal bases [4], we consider the q → t ). We find that for t = p − with p an odd prime,this limit is proportional to a count of certain chains of groups of p -adic type. While it iswell-known that Hall-Littlewood polynomials are intimately related to p -adic representationtheory, it is somewhat surprising to find p -adic quantities arising in the context of quantumgroups.We will now state our results more precisely. First recall that a Gelfand-Tsetlin patternof shape λ ∈ P ( n )+ is a sequence λ = λ (0) (cid:23) λ (1) (cid:23) · · · (cid:23) λ ( n − , where λ ( i ) ∈ P ( n − i )+ and (cid:23) denotes the interlacing relation: λ ( i +1) j − λ ( i ) j +1 ∈ Z + , λ ( i ) j +1 − λ ( i +1) j +1 ∈ Z + . This may bevisualized as an array consisting of n rows with the parts of λ in the first row, parts of λ (1) in the second row, etc. There is a canonical basis of V λ which is indexed by GT ( λ ), theGelfand-Tsetlin patterns of shape λ .Our aim is to compute the expansion of the trace function Φ ( k ) in the Gelfand-Tsetlinbasis for V λ +( k − ρ n . We would like to index these patterns in a uniform way with respectto the parameter k ∈ N . Conveniently, there is a canonical way of doing this for theGelfand-Tsetlin patterns whose coefficient in the expansion of Φ ( k ) is non-zero: Definition 1.2.
Let λ ∈ P ( n )+ , and let λ = µ (0) ⊃ µ (1) ⊃ · · · ⊃ µ ( n − be such that: (1) µ ( i ) ∈ P ( n − i )+ . (2) µ ( i ) j − µ ( i +1) j ∈ Z + , ≤ j ≤ n − i − . (3) µ ( i ) j − µ ( i +1) j − ≤ k − , ≤ j ≤ n − i .Define µ ( i ) = µ ( i ) + ( k − ρ n − i + ( k − · ( i/ , . . . , i/ , then ( µ (0) (cid:23) µ (1) (cid:23) · · · (cid:23) µ ( n − ) is a Gelfand-Tsetlin pattern of shape λ + ( k − ρ n . We can now give our formula for the coefficients in the expansion of Φ ( k ) λ ( x ) in theGelfand-Tsetlin basis, along with a new branching formula for the functions e Φ( x ; q, t ). Thiswill be expressed in terms of functions ψ γ/δ , Ω γ/δ , which are plethystic substitutions ofskew Macdonald polynomials, and d α , which is the norm with respect to a particular innerproduct, see the Background section for more details. N THE EXPANSION OF CERTAIN VECTOR-VALUED CHARACTERS OF U q ( gl n ) 3 Theorem 1.3.
For µ ⊂ λ with λ ∈ P ( n )+ , µ ∈ P ( n − , define (2) c λ,µ ( q, t ) = d µ ( q , t ) X β ∈P ( n − µ ⊆ β (cid:22) λ ψ λ/β ( q , t ) d β ( q , t ) Ω β/µ ( q , t ) . Then the trace functions e Φ( x ; q, t ) satisfy the branching rule: e Φ ( n ) λ ( x ; q, t ) = X µ ∈P ( n +1)+ µ ⊂ λ c λ,µ ( q, t ) · e Φ ( n − µ ( x ; q, t ) · x ρ ( λ,µ ) n , where ρ ( λ, µ ) = n − X i =1 λ i − µ i .Moreover, with respect to the Gelfand-Tsetlin basis of V λ +( k − ρ n , the diagonal coefficientof the intertwining operator φ ( k ) corresponding to the Gelfand-Tsetlin pattern Λ = ( λ (0) (cid:23)· · · (cid:23) λ ( n − ) ∈ GT ( λ + ( k − ρ n ) is equal to c Λ ( q, q k ) = Y ≤ i ≤ n − c µ ( i − ,µ ( i ) ( q, q k ) , ∃ ( µ (0) ⊇ · · · ⊇ µ ( n − ) s.t. λ ( i ) = µ ( i ) , otherwise . Kashiwara’s crystal bases [4] allow one to interpret finite-dimensional representations of U q ( gl n ) in the “crystal limit” q →
0. Remarkably, there is a rich combinatorial structurein the crystal limit. Since the Gelfand-Tsetlin basis yields a crystal basis for V λ , it seemsnatural to consider the limit as q → Theorem 1.4.
Let µ ⊂ λ with λ ∈ P ( n )+ , µ ∈ P ( n − , then we have (3) lim q → c λ,µ ( q, t ) = b µ ( t ) b λ ( t ) (1 − t ) sk λ/µ ( t ) = t P j ( λ ′ j − µ ′ j ) Y j ≥ (cid:18) λ ′ j − µ ′ j +1 λ ′ j − λ ′ j +1 (cid:19) t . Here the coefficients sk λ/µ ( t ) are those studied in [12, 7] in the context of Pieri rules. Notethat when t = p − for an odd prime p , sk λ/µ ( t ) = t n ( λ ) − n ( µ ) α λ ( µ ; p ) , where α λ ( µ ; p ) is the number of subgroups of type µ in a finite abelian p -group of type λ .For S = ( µ (0) ⊃ µ (1) ⊃ · · · ⊃ µ ( n − ) with µ ( i ) ∈ P ( n − i )+ , we define the coefficient(4) sk S ( t ) = sk µ (0) /µ (1) ( t ) sk µ (1) /µ (2) ( t ) · · · sk µ ( n − /µ ( n − ( t ) . Note that when t = p − , sk S ( t ) is (up to a power of t ) the number of nested chains ofsubgroups with types specified by the sequence S . We also let wt ( S ) = (cid:0) ρ ( µ ( n − , , ρ ( µ ( n − , µ ( n − ) , · · · , ρ ( µ (1) , µ (2) ) , ρ ( µ (0) , µ (1) ) (cid:1) . Theorem 1.5.
Let λ ∈ P ( n )+ . Then (5) lim q → e Φ λ ( x ; q, t ) = (1 − t ) n b λ ( t ) X S =( λ = µ (0) ⊃ µ (1) ⊃···⊃ µ ( n − ) µ ( i ) ∈P ( n − i )+ sk S ( t ) x wt ( S ) . Note that the coefficients for the Gelfand-Tsetlin basis of V λ +( k − ρ n are obtained byspecializing t = q k in (2), and hence we are only able to obtain the t = 0 specialization of(3) in the crystal limit. As mentioned above, there is a representation theoretic realizationof e Φ( x ; q, t ), with t algebraically independent from q , as the trace function of an intertwinerbetween infinite-dimensional modules over C ( t ) ⊗ U q ( gl n ). There is an analogue of theGelfand-Tsetlin basis for these modules, and we can obtain (3) for general t as the q → VIDYA VENKATESWARAN limit of the coefficients in the expansion of e Φ( x ; q, t ) with respect to this basis (see Section 5within the paper for more details about this). Unfortunately these modules do not fit intoKashiwara’s framework, and so we have not been able to find a direct interpretation of (3)in terms of crystal bases. However, the simple combinatorial structure of our formula in thelimit q → Acknowledgements.
The author would like to thank Pavel Etingof for suggesting thiswork, and for many helpful discussions and comments. She would also like to thank EricRains and Ole Warnaar for helpful comments.2.
Background on symmetric function theory
Recall that λ = ( λ , . . . , λ n ) ∈ ( Z + ) n is a partition if λ i ≥ λ i +1 . There is a partial orderon partitions defined by λ > µ if and only if P λ i = P µ i and for some k < n we have λ i = µ i for all i ≤ k and λ k +1 > µ k +1 . We will work with polynomials of n variables, i.e.,over C [ x , . . . , x n ]. For λ ∈ Z n , we let x λ = x λ · · · x λ n n .We fix k ∈ N , and set t = q k . Let ρ = ( n − , n − , . . . , − n ) be half the sum of the positiveroots; we will also write ρ n when it is not clear from context. Note that(6) ρ n − ρ n − = (cid:16) n − , n − , . . . , − n (cid:17) − (cid:16) n − , n − , . . . , − n (cid:17) = (cid:16) , . . . , , − n (cid:17) . We now define a number of different coefficients arising from symmetric function theory,see [10]; we also review some relevant results from the literature.
Definition 2.1.
Define the functions g ( γ ; q , t ) for γ a partition by g ( γ ; q , t ) = t | γ | ( t − q ; q ) γ · · · ( t − q ; q ) γ n − ( q ; q ) γ · · · ( q ; q ) γ n − . Definition 2.2.
Let c δγµ ( q, t ) be the coefficients in the following Pieri rule: m ( n ) γ ( x ) P ( n ) µ ( x ; q, t ) = X δ c δγµ ( q, t ) P ( n ) δ ( x ; q, t ) . We note that the coefficients c δγµ can be determined via the change of basis coefficients { m ( n ) γ ( x ) } → { P ( n ) η ( x ; q, t ) } in conjunction with the Pieri coefficients that express the prod-uct P ( n ) η ( x ; q, t ) P ( n ) µ ( x ; q, t ) in the Macdonald polynomial basis. Definition 2.3.
Let Ω β/µ ( q , q k ) be the coefficient on P ( n − β ( x ; q , q k ) in the expansionof P ( n − µ ( x ; q , q k ) n − Y i =1 ( q x i ; q ) ∞ ( q k x i ; q ) ∞ in the basis { P ( n − β ( x ; q , q k ) } β . We now recall the branching rule for Macdonald polynomials.
Theorem 2.4. P ( n ) λ ( x ; q, t ) = X µ (cid:22) λ x | λ − µ | n ψ λ/µ ( q, t ) P ( n − µ ( x ; q, t ) Proof.
See (1.7) of [8] for example. (cid:3)
Remark.
There is a product formula for the coefficients ψ λ/µ ( q, t ) appearing above ( [10] p342) ψ λ/µ ( q, t ) = Y ≤ i ≤ j ≤ l ( µ ) f ( q µ i − µ j t j − i ) f ( q λ i − λ j +1 t j − i ) f ( q λ i − µ j t j − i ) f ( q µ i − λ j +1 t j − i ) , N THE EXPANSION OF CERTAIN VECTOR-VALUED CHARACTERS OF U q ( gl n ) 5 where f ( a ) = ( at ) ∞ / ( aq ) ∞ with ( a ) ∞ = Q i ≥ (1 − aq i ) . Proposition 2.5.
We have lim q → ψ λ/µ ( q, t ) = Y { j : λ ′ j = µ ′ j and λ ′ j +1 = µ ′ j +1 +1 } (1 − t m j ( µ ) ) if λ/µ is a horizontal strip, and zero otherwise.Proof. This follows from the branching rule for Hall-Littlewood polynomials (see for example[10] p228 (5.5’), (5.14’)). (cid:3)
Definition 2.6.
Let φ λ/µ ( t ) = Y { j : λ ′ j = µ ′ j +1 and λ ′ j +1 = µ ′ j +1 } (1 − t m j ( λ ) ) , if λ/µ is a horizontal strip, and zero otherwise. Note that these coefficients are the q → φ λ/µ ( q, t ) which also arise asbranching coefficients. Remark.
The functions φ λ/β ( q, t ) , Ω β/ ¯ µ ( q, t ) have interpretations in terms of skew Mac-donald polynomials (in parameters q, t ) with plethystic substitutions. In particular, we have φ λ/β ( q, t ) = Q λ/β (1) and Ω β/ ¯ µ ( q, t ) = Q β/ ¯ µ (cid:0) t − q − t (cid:1) = t | β/ ¯ µ | Q β/ ¯ µ (cid:0) − q/t − t (cid:1) (see for example [12] ), and both these quantities have nice factorized forms. We will write ψ λ/µ ( t ), g ( γ ; t ), etc. to denote the limit q → Definition 2.7. [5, 7]
For any skew shape λ/µ , define the coefficients sk λ/µ ( t ) = t P j ( λ ′ j − µ ′ j ) Y j ≥ (cid:18) λ ′ j − µ ′ j +1 m j ( µ ) (cid:19) t . Theorem 2.8. [5, 7]
For a partition λ and r ≥ , we have P ( n ) λ ( x ; t ) s ( n ) r ( x ) = X λ + sk λ + /λ ( t ) P ( n ) λ + ( x ; t ) , with the sum over partitions λ ⊂ λ + for which | λ + /λ | = r . We now recall two inner products that will appear throughout the paper. We let h· , ·i n denote the Macdonald inner product (defined via integration over the n -torus). In particular,(7) h P ( n ) λ ( x ; q, t ) , P ( n ) µ ( x ; q, t ) i n = Z T n P ( n ) λ ( x ; q, t ) P ( n ) µ ( x − ; q, t ) ˜∆ S ( x ; q, t ) dT = δ λ,µ d λ ( q, t ) , where an explicit formula for d λ ( q, t ) can be found in [10]. Also let Q ( n ) µ ( x ; q, t ) = b µ ( q, t ) P ( n ) µ ( x ; q, t )be scalar multiples of the Macdonald polynomials, and recall the other inner product h· , ·i ′ which satisfies(8) h P ( n ) λ ( x ; q, t ) , Q ( n ) µ ( x ; q, t ) i ′ = δ λ,µ (so that h P ( n ) λ ( x ; q, t ) , P ( n ) λ ( x ; q, t ) i ′ = b λ ( q,t ) ). Note that this inner product is independentof n , and we have lim n →∞ h· , ·i n = h· , ·i ′ . We have b λ ( t ) = Y i ≥ φ m i ( λ ) ( t ) , where m i ( λ ) denotes the number of times i occurs as a part of λ and φ r ( t ) = (1 − t )(1 − t ) · · · (1 − t r ) . VIDYA VENKATESWARAN
We also have d λ ( t ) = 1(1 − t ) n Y i ≥ φ m i ( λ ) ( t ) , so that if l ( λ ) = n , b λ ( t )(1 − t ) n = d λ ( t ).We recall the following fact relating the branching coefficients φ λ/β and ψ λ/β [10]. Proposition 2.9.
We have φ λ/β ( q, t ) /b λ ( q, t ) = ψ λ/β ( q, t ) /b β ( q, t ) . Note that, using (7) and (8), the coefficients c δλµ and sk λ + /λ may be defined in terms ofinner products. We have c δλµ ( q, t ) = h m ( n ) γ ( x ) P ( n ) µ ( x ; q, t ) , Q ( n ) δ ( x ; q, t ) i ′ = d δ ( q, t ) h m ( n ) γ ( x ) P ( n ) µ ( x ; q, t ) , P ( n ) δ ( x ; q, t ) i and similarly h P ( n ) λ ( x ; t ) s ( n ) r ( x ) , Q ( n ) λ + ( x ; t ) i ′ = sk λ + /λ ( t ) . The Gelfand-Tsetlin basis expansion
In this section, we fix k ∈ N and set t = q k . We will prove Theorem 1.3 of the introduction.Namely, we will expand the trace function Φ ( n ) λ ( x ) in the Gelfand-Tsetlin basis and computethe diagonal coefficients c Λ ( q, t ). We will use the multiplicity-one decomposition of V λ +( k − ρ as a U q ( gl n − )-module, and iterate, in order to do this.Etingof and Kirillov [2] provide the following closed form for the trace function at λ = 0: Proposition 3.1. Φ ( n )0 ( x ) = k − Y i =1 Y α ∈ R + ( x α/ − q i x − α/ ) = x ( k − ρ k − Y i =1 Y n ≥ l>m ≥ (1 − q i x l /x m ) Proposition 3.2. φ ( n )0 ( x ) φ ( n − ( x ) = x ( k − ρ n − ρ n − ) ∞ X l ,...,l n − =0 t P l i ( t − q ; q ) l · · · ( t − q ; q ) l n − ( q ; q ) l · · · ( q ; q ) l n − x P l i n x l · · · x l n − n − Proof.
By Proposition 3.1, we have φ ( n )0 ( x ) φ ( n − ( x ) = x ( k − ρ n − ρ n − ) k − Y i =1 n − Y j =1 (1 − q i x n /x j ) . Now note that, for fixed 1 ≤ j ≤ n − k − Y i =1 (1 − q i x n /x j ) = ( x n /x j ; q ) k ( x n /x j ; q ) = ( x n /x j ; q ) ∞ ( q k x n /x j ; q ) ∞ · ( q x n /x j ; q ) ∞ ( x n /x j ; q ) ∞ = ( q x n /x j ; q ) ∞ ( q k x n /x j ; q ) ∞ . Now, putting t = q k and using the q -binomial theorem,( q x n /x j ; q ) ∞ ( t x n /x j ; q ) ∞ = ∞ X m =0 ( t − q ; q ) m ( q ; q ) m ( t x n /x j ) m . Taking the product over all 1 ≤ j ≤ n − x ( k − ρ n − ρ n − ) gives theresult. (cid:3) N THE EXPANSION OF CERTAIN VECTOR-VALUED CHARACTERS OF U q ( gl n ) 7 Lemma 3.3.
Let λ be fixed with l ( λ ) = n . Then the map µ → ¯ µ = µ + ( k − (cid:16) ρ n − + (cid:16) , , . . . , (cid:17)(cid:17) = µ + ( k − ρ n | n − is a bijection between: • µ ⊂ λ , such that λ j +1 − µ j ≤ k − for all j • ¯ µ (cid:22) λ + ( k − ρ n , such that ¯ µ − ( k − ρ n | n − ∈ P + Proof.
Follows from the definition of the interlacing condition and Equation 6. (cid:3)
Proposition 3.4.
The following branching rule for trace functions holds: Φ ( n ) λ ( x ; q, q k ) = ( x · · · x n − ) k − X µ ⊂ λ x ρ ( λ,µ ) n a λ,µ ( q )Φ ( n − µ ( x ; q, q k ) for some coefficients a λ,µ ( q ) .Proof. One first notes the multiplicity-free decomposition of V λ +( k − ρ n as a module over U q ( gl n − ): V λ +( k − ρ n | U q ( gl n − ) = ⊕ ¯ µ (cid:22) λ +( k − ρ n V ¯ µ . Thus, we haveΦ λ ( x ; q, q k ) = Tr( φ λ · x h ) = X ¯ µ (cid:22) λ +( k − ρ n Tr( φ λ · x h | V ¯ µ )= X ¯ µ (cid:22) λ +( k − ρ n Tr (cid:16) (Proj V ¯ µ ⊗ Id ) ◦ φ λ ◦ x h | V ¯ µ (cid:17) , since trace only takes into account diagonal coefficients.We have φ λ | V ¯ µ : V ¯ µ → V λ +( k − ρ n ⊗ U ≃ ⊕ α (cid:22) λ +( k − ρ n V α ⊗ U, thus (Proj V ¯ µ ⊗ Id ) ◦ φ λ | V ¯ µ : V ¯ µ → V ¯ µ ⊗ U is an intertwining operator. By [2], this implies that(Proj V ¯ µ ⊗ Id ) ◦ φ λ | V ¯ µ = ( ˆ a λ, ¯ µ ( q ) · φ ¯ µ − ( k − ρ n − , if ¯ µ − ( k − ρ n − ∈ P + , else,for some coefficients ˆ a λ, ¯ µ ( q ). Thus, we have,Φ λ ( x ; q, q k ) = X ¯ µ (cid:22) λ +( k − ρ n ¯ µ − ( k − ρ n − ∈P + ˆ a λ, ¯ µ ( q ) · x ρ ( λ, ¯ µ ) n · Φ ¯ µ − ( k − ρ n − ( x ; q, q k ) . Finally we reparametrize by setting µ = ¯ µ − ( k − ρ n − − ( k − ) n − and defining a λ,µ ( q ) = ˆ a λ, ¯ µ ( q ) with the condition that a λ,µ = 0 if λ j +1 − µ j ≤ k − j . The result now follows by the previous Lemma. (cid:3) By iterating the branching rule of the previous propostion and recalling that the Gelfand-Tsetlin basis is also obtained by iterating the multiplicty-free decomposition, we obtain thefollowing result.
Proposition 3.5.
We have the following formula for Φ ( n ) λ ( x ; q, q k ) as a sum over Gelfand-Tsetlin patterns ( λ (0) , λ (1) , . . . , λ ( n − ) with λ (0) = λ + ( k − ρ n : Φ ( n ) λ ( x ; q, q k ) = X Λ ∈ GT ( λ +( k − ρ n )Λ=( λ (0) , ··· λ ( n − ) Y ≤ i ≤ n − a λ ( i − ,λ ( i ) ( q ) x wt (Λ) VIDYA VENKATESWARAN We will show that the coefficients a λ,µ ( q ) are equal to c λ,µ ( q, q k ) defined in the intro-duction. We will prove this through a series of propositions. Recall the definitions of thefunctions g ( · ; · , · ) , ψ · / · ( · , · ) , c ·· , · ( · ; · ) in the introduction. Lemma 3.6.
For any m ∈ C , the branching coefficients a λ,µ ( q ) satisfy the shift invariance: a λ + m n ,µ + m n − ( q ) = a λ,µ ( q ) . Proof.
The intertwining operator φ ( k ) , as well as the multiplicity one decomposition used inthe proof of Proposition 3.4, is determined by the U q ( sl n )-module structure. Indeed, U q ( gl n )differs only from U q ( sl n ) by the addition of the central element q ǫ + ··· + ǫ n . The result thenfollows easily from the observation that, as U q ( sl n )-modules, V λ + m n is isomorphic to V λ forany partition λ and any m ∈ C . (cid:3) Remark.
By the previous Lemma, to compute a λ,µ ( q ) for λ ∈ P ( n )+ , µ ∈ P ( n − , we mayassume that λ, µ are partitions with l ( λ ) = n , l ( µ ) = n − . We will make this assumptionimplicitly throughout the paper. Proposition 3.7.
The branching coefficients satisfy the following formula: a λ,µ ( q ) = X β (cid:22) λ,l ( β ) ≤ n − γ ∈ Z n − ≥ a partition g ( γ ; q , q k ) ψ λ/β ( q , q k ) c µ +( k − n − − γ +( k − n − ,β ( q , q k ) . Proof.
Combining Theorem 1.1 with Proposition 3.4 gives the following: φ ( n )0 ( x ; q, q k ) φ ( n − ( x ; q, q k ) P ( n ) λ ( x ; q , q k ) = X µ ⊂ λ x ρ ( λ,µ ) n a λ,µ ( q ) P ( n − µ +( k − ) n − ( x ; q , q k ) . We then use Theorem 2.4 to rewrite this as(9) φ ( n )0 ( x ; q, q k ) φ ( n − ( x ; q, q k ) X µ (cid:22) λl ( µ ) ≤ n − x | λ − µ | n ψ λ/µ ( q , q k ) P ( n − µ ( x ; q , q k )= X µ ⊂ λ x ρ ( λ,µ ) n a λ,µ ( q ) P ( n − µ +( k − ) n − ( x ; q , q k ) . Now note from Proposition 3.2, we have φ ( n )0 ( x ; q, q k ) φ ( n − ( x ; q, q k ) = x ( k − ρ n − ρ n − ) ×× X γ ∈ Z n − ≥ q k ( P i γ i ) ( q − k − ; q ) γ · · · ( q − k − ; q ) γ n − ( q ; q ) γ · · · ( q ; q ) γ n − x ( P i γ i ) n m − γ ( x , . . . , x n − ) , where the sum is over γ = ( γ , . . . , γ n − ) a partition. Now recall that ( k − ρ n − ρ n − ) =( k − , , . . . , , − n ) ∈ Z n , so we have φ ( n )0 ( x ; q, q k ) φ ( n − ( x ; q, q k ) = X γ ∈ Z n − ≥ a partition x ( k − − n )+ | γ | n g ( γ ; q , q k ) m − γ +( k − ,..., ) ( x , . . . , x n − ) , where we have used Definition (2.1).We use the previous equation, along with (9), and multiply both sides by the monomial( x · · · x n − ) ( k − ,..., ) to obtain the equation: N THE EXPANSION OF CERTAIN VECTOR-VALUED CHARACTERS OF U q ( gl n ) 9 X γ ∈ Z n − ≥ a partition µ (cid:22) λl ( µ ) ≤ n − x | λ − µ | +( k − − n )+ | γ | n g ( γ ; q , q k ) ψ λ/µ ( q , q k ) ×× m − γ +( k − n − ( x , . . . , x n − ) P ( n − µ ( x ; q , q k )= X µ ⊂ λ x ρ ( λ,µ ) n a λ,µ ( q ) P ( n − µ +( k − n − ( x ; q , q k ) . Next we use Definition (2.2) to rewrite this as X γ ∈ Z n − ≥ a partition µ (cid:22) λ,l ( µ ) ≤ n − δ a partition x | λ − µ | +( k − − n )+ | γ | n g ( γ ; q , q k ) ψ λ/µ ( q , q k ) ×× c δ − γ +( k − n − ,µ ( q , q k ) P ( n − δ ( x ; q , q k )= X µ ⊂ λ x ρ ( λ,µ ) n a λ,µ ( q ) P ( n − µ +( k − n − ( x ; q , q k ) . Since both LHS and RHS are expansions in the Macdonald polynomial basis, the corre-sponding coefficients must be equal. That is, a λ,µ ( q ) = X β (cid:22) λ,l ( β ) ≤ n − γ ∈ Z n − ≥ a partition g ( γ ; q , q k ) ψ λ/β ( q , q k ) c µ +( k − n − − γ +( k − n − ,β ( q , q k ) , as desired. (cid:3) Proposition 3.8.
Let µ ⊂ λ with l ( µ ) ≤ n − . Then we have a λ,µ ( q ) = X β (cid:22) λ,l ( β ) ≤ n − l ( γ ) ≤ n − g ( γ ; q , q k ) ψ λ/β ( q , q k ) d µ ( q , q k ) d β ( q , q k ) c βγ,µ ( q , q k ) . Proof.
Using standard facts about integration over T n , we have c µ +( k − n − − γ +( k − n − ,β ( q , q k )= h m − γ +( k − n − ( x ) P β ( x ; q , q k ) , P µ +( k − n − ( x ; q , q k ) i d µ +( k − n − ( q , q k )= h m − γ ( x ) P β ( x ; q , q k ) , P µ ( x ; q , q k ) i d µ +( k − n − ( q , q k )= d µ +( k − n − ( q , q k ) h m γ ( x ) P µ ( x ; q , q k ) , P β ( x ; q , q k ) i = d µ ( q , q k ) d β ( q , q k ) c βγ,µ ( q , q k ) , where we have used d µ +( k − n − ( q , q k ) = d µ ( q , q k ), which follows from the definition of h· , ·i . Combining this with the previous theorem gives the result. (cid:3) We are now prepared to provide a proof of Theorem 1.3, mentioned in the introductionto this paper. The proof relies on the previous propositions proved in this section.
Proof of Theorem 1.3.
By the previous proposition, we have a λ,µ ( q ) = d µ ( q , q k ) X β (cid:22) λ,l ( β ) ≤ n − l ( γ ) ≤ n − g ( γ ; q , q k ) ψ λ/β ( q , q k ) 1 d β ( q , q k ) c βγ,µ ( q , q k ) . Now, note that for fixed β , we have X l ( γ ) ≤ n − g ( γ ; q , q k ) c βγ,µ ( q , q k )= X l ( γ ) ≤ n − q k | γ | ( q − k q ; q ) γ · · · ( q − k q ; q ) γ n − ( q ; q ) γ · · · ( q ; q ) γ n − c βγ,µ ( q , q k )= X l ( γ ) ≤ n − q k | γ | ( q − k q ; q ) γ · · · ( q − k q ; q ) γ n − ( q ; q ) γ · · · ( q ; q ) γ n − h m ( n − γ ( x ) P ( n − µ ( x ; q , q k ) , Q ( n − β ( x ; q , q k ) i ′ = D(cid:16) X l ( γ ) ≤ n − q k | γ | ( q − k q ; q ) γ · · · ( q − k q ; q ) γ n − ( q ; q ) γ · · · ( q ; q ) γ n − m ( n − γ ( x ) (cid:17) P ( n − µ ( x ; q , q k ) , Q ( n − β ( x ; q , q k ) E ′ . Now recall the following identity [10, p314]: g n ( x ; q , t ) = X | µ | = n ( t ; q ) µ ( q ; q ) µ m µ ( x ) . Using this, we may write the previous equation as X l ( γ ) ≤ n − g ( γ ; q , q k ) c βγ, ¯ µ ( q , q k )= D(cid:16) X r ≥ q kr g ( n − r ( x ; q , q − k q ) (cid:17) P ( n − µ ( x ; q , q k ) , Q ( n − β ( x ; q , q k ) E ′ . Also note [10, p311] that we have the generating function identity (for arbitrary q, t, y ): X n ≥ g n ( x ; q, t ) y n = Y i ≥ ( tx i y ; q ) ∞ ( x i y ; q ) ∞ ;thus we have X l ( γ ) ≤ n − g ( γ ; q , q k ) c βγ,µ ( q , q k )= D Y i ≥ ( q x i ; q ) ∞ ( q k x i ; q ) ∞ P ( n − µ ( x ; q , q k ) , Q ( n − β ( x ; q , q k ) E ′ . Combining this with the original sum yields a λ,µ ( q ) = d µ ( q , q k ) ×× X β (cid:22) λl ( β ) ≤ n − ψ λ/β ( q , q k ) d β ( q , q k ) D Y i ≥ ( q x i ; q ) ∞ ( q k x i ; q ) ∞ P ( n − µ ( x ; q , q k ) , Q ( n − β ( x ; q , q k ) E ′ = d µ ( q , q k ) X β (cid:22) λl ( β ) ≤ n − ψ λ/β ( q , q k ) d β ( q , q k ) Ω β/µ ( q , q k )by definition of Ω β/µ ( q , q k ). But this is exactly equal to c λ,µ ( q, q k ) as defined in ((2)). (cid:3) Remarks.
The coefficients c λ,µ ( q, t ) do not appear to factor nicely at the q -level, due tothe restriction on length in the sum. For example, for λ = (2 , and µ = (1) one obtains (1 − t )(1 − qt )(1 − q − q + t ) / (1 − qt ) , and the term (1 − q − q + t ) cannot be expressedas a product of (1 − q i t j ) . N THE EXPANSION OF CERTAIN VECTOR-VALUED CHARACTERS OF U q ( gl n ) 11 The q → limit We will look at the q → c λ,µ ( q, t ). We find that the formula hasa nice product form, in terms of certain p -adic counts. The simplification of these coefficientsat q = 0 may be related to the crystal basis structure of the Gelfand-Tsetlin basis, althoughwe have not investigated a direct link.We note that the parameters q, t are linked in the finite-dimensional case since t = q k and k ∈ N , so we cannot take q → t → t as a formal variable and in Section 5, we will relate it to the representationtheory by using Verma modules.The goal of this section is to prove Theorems 1.4 and 1.5 mentioned in the introduction. Definition 4.1.
We let c λ,µ ( t ) denote lim q → c λ,µ ( q, t ) . Theorem 4.2.
Let λ be a partition of length n , and µ ⊂ λ, µ ∈ P + . Then retaining thenotation of the previous sections, we have c λ,µ ( t ) = b µ ( t ) b λ ( t ) X β (cid:22) λl ( β ) ≤ n − φ λ/β ( t ) t | β/µ | sk β/µ ( t ) . Proof.
We use Theorem 1.3, the functions there admit the limit q →
0. We also use thatΩ β/µ ( q, t ) = Q β/µ (cid:16) t − q − t (cid:17) = t | β/µ | Q β/µ (cid:16) − q − t (cid:17) and lim q → Q β/µ (cid:16) − q − t (cid:17) = sk β/µ ( t ) , where the skew Macdonald polynomials are taken with respect to the parameters ( q, t ). Thisgives the following c λ,µ ( t ) = d µ ( t ) X β (cid:22) λl ( β ) ≤ n − ψ λ/β ( t ) d β ( t ) t | β/µ | sk β/µ ( t ) . Finally, one notes that d β ( t ) = b β ( t ) and d µ ( t ) = b µ ( t ) , and by the q → φ λ/β ( t ) /b λ ( t ) = ψ λ/β ( t ) /b β ( t );using this in the previous equation gives the result. (cid:3) Our next goal is to obtain a nice factorized product form for c λ,µ ( t ). We use a q → q -Pfaff-Saalsch¨utz formula: Theorem 4.3 ([11, Corollary 4.9]) . Let µ ⊂ λ be partitions, then for arbitrary parameters a, b, c we have the following identity: X β ( a ) β ( c ) β Q λ/β (cid:18) a − b − t (cid:19) Q β/µ (cid:18) b − c − t (cid:19) = ( a ) µ ( b ) λ ( b ) µ ( c ) λ Q λ/µ (cid:18) a − c − t (cid:19) . Proposition 4.4.
Let λ be a partition of length n and let µ ≤ λ with l ( µ ) = n − . Then (1) X β (cid:22) λ φ λ/β ( t ) t | β/µ | sk β/µ ( t ) = sk λ/µ ( t )(2) X β (cid:22) λl ( β ) ≤ n − φ λ/β ( t ) t | β/µ | sk β/µ ( t ) = (1 − t ) · sk λ/µ ( t ) Proof.
Take a = q , b = qt , c = q in in Theorem 4.3: X β ( q ) β ( q ) β Q λ/β (cid:18) q − qt − t (cid:19) Q β/µ (cid:18) qt − q − t (cid:19) = ( q ) µ ( qt ) λ ( qt ) µ ( q ) λ Q λ/µ (cid:18) q − q − t (cid:19) . Using the relation Q λ/µ (cid:0) aq − bq − t (cid:1) = q | λ/µ | Q λ/µ (cid:0) a − b − t (cid:1) , we have X β ( q ) β ( q ) β Q λ/β (cid:0) (cid:1) Q β/µ (cid:18) t − q − t (cid:19) = ( q ) µ ( qt ) λ ( qt ) µ ( q ) λ Q λ/µ (cid:18) − q − t (cid:19) . (1) then follows by taking the limit q → X β (cid:22) λl ( β )= n φ λ/β ( t ) t | β/µ | sk β/µ ( t ) = t · sk λ/µ ( t ) . Reindex the sum by replacing β = β ′ + 1 n . Then we have X β ′ (cid:22) λ − n φ λ/ ( β ′ +1 n ) ( t ) t | β ′ /µ | t n sk ( β ′ +1 n ) /µ ( t )We have the following two identities: φ λ/ ( β ′ +1 n ) ( t ) = φ ( λ − n ) /β ′ ( t ) sk ( β ′ +1 n ) /µ ( t ) = sk β ′ / ( µ − ( n − ) ( t ) , which can be seen by using the explicit formulas in Section 2. It follows that X β ′ (cid:22) ( λ − n ) φ ( λ − n ) /β ′ ( t ) t | β ′ / ( µ − n − ) | t sk β ′ / ( µ − ( n − ) ( t ) = t · sk ( λ − n ) / ( µ − ( n − ) ( t ) . Applying the identity above again completes the proof. (cid:3)
We now provide a proof of Theorem 1.4, mentioned in the introduction. The proof relieson the previous results of this section.
Proof of Theorem 1.4.
By Proposition 4.4, we have c λ,µ ( t ) = b µ ( t ) b λ ( t ) X β (cid:22) λl ( β ) ≤ n − φ λ/β ( t ) t | β/µ | sk β/µ ( t ) = b µ ( t ) b λ ( t ) (1 − t ) sk λ/µ ( t ) , which gives the first equality. By the definitions of b λ ( t ) , sk λ/µ ( t ), this is equal to Q i ≥ φ m i ( µ ) ( t ) Q i ≥ φ m i ( λ ) ( t ) (1 − t ) t P j ( λ ′ j − µ ′ j ) Y j ≥ (cid:18) λ ′ j − µ ′ j +1 µ ′ j − µ ′ j +1 (cid:19) t = (1 − t ) t P j ( λ ′ j − µ ′ j ) Y j ≥ φ λ ′ j − µ ′ j +1 ( t ) φ λ ′ j − λ ′ j +1 ( t ) φ λ ′ j − µ ′ j ( t )= (1 − t ) φ λ ′ − µ ′ ( t ) t P j ( λ ′ j − µ ′ j ) Y j ≥ φ λ ′ j − µ ′ j +1 ( t ) φ λ ′ j − λ ′ j +1 ( t ) φ λ ′ j +1 − µ ′ j +1 ( t )= t P j ( λ ′ j − µ ′ j ) Y j ≥ (cid:18) λ ′ j − µ ′ j +1 λ ′ j − λ ′ j +1 (cid:19) t , where we have used m i ( µ ) = µ ′ i − µ ′ i +1 and λ ′ − µ ′ (because l ( λ ) = n and l ( µ ) = n − (cid:3) N THE EXPANSION OF CERTAIN VECTOR-VALUED CHARACTERS OF U q ( gl n ) 13 We recall that, as mentioned in the introduction, there is a p -adic interpretation forcoefficients sk λ/µ ( t ) and thus for c λ,µ ( t ). More precisely, sk λ/µ ( t ) = t n ( λ ) − n ( µ ) α λ ( µ ; t − );where α λ ( µ ; p ) is the number of subgroups of type µ in a finite abelian p -group of type λ ,see [12] for example, and the references therein. Proof of Theorem 1.5.
Recall that for S = ( µ (0) ⊃ µ (1) ⊃ · · · ⊃ µ ( n − ) with µ ( i ) ∈ P ( n − i )+ ,we defined the coefficient sk S ( t ) as a product of sk µ ( i − /µ ( i ) ( t ) in (4). By Definition 1.2,one can associate to S a Gelfand-Tsetlin array Λ. Thus, using Theorem 1.4, we havelim q → c Λ ( q, t ) = (1 − t ) n b λ ( t ) sk S ( t ) . Using this along with Theorem 1.3 gives the result. (cid:3)
Note that when t = p − for p an odd prime, the coefficients appearing in both Theorems1.4 and 1.5 are explicit p -adic counts. Corollary 4.5.
Let λ be a partition. We have the following formula for the Hall-Littlewoodpolynomial: P λ ( x , . . . , x n ; t ) = 1 b λ ( t ) X S =( λ = µ (0) ⊃ µ (1) ⊃···⊃ µ ( n − ) µ ( i ) ∈P ( n − i )+ sk S ( t ) x wt ( S ) X S ′ =(0 n = µ (0) ⊃ µ (1) ⊃···⊃ µ ( n − ) µ ( i ) ∈P ( n − i )+ sk S ′ ( t ) x wt ( S ′ ) . Proof.
Follows from Theorem 1.1 along with Theorem 1.5. (cid:3) Verma modules and algebraically independent t We have computed the expansion of the Macdonald vector-valued characters Φ ( k ) λ ( x ; q )with respect to the Gelfand-Tsetlin basis of V λ +( k − ρ . These are expressed in terms ofrational functions in q, t which appear naturally in symmetric function theory, specializedto t = q k . In the previous section we showed that, for algebraically independent t , thesecoefficients admit a simple limit as q →
0, which is related to natural quantities appearing in p -adic representation theory. Note however that in the representation theoretic realizationof Φ ( k ) λ we have t = q k , and hence we can only obtain the t = 0 specialization of our formulain the q → ( k ) λ to algebraically independent t by replacing the finite-dimensional irreducible module V λ +( k − ρ by a suitable infinitedimensional irreducible Verma module. In this section we outline their construction, whichallows us to obtain a representation theoretic realization of our formula for algebraicallyindependent t .Consider the algebra C ( t ) ⊗ U q ( gl n ), i.e. the quantum group U q ( gl n ) where the coefficientfield is expanded to C ( t ) ⊗ C ( q ) (note this can be identified with the subalgebra of C ( q, t )spanned by products of the form r ( q ) · r ( t ) for rational functions r , r ). We have thefollowing analogues of the finite-dimensional modules V λ +( k − ρ : Definition 5.1.
For λ ∈ P ( n ) , the module M λ,t over C ( t ) ⊗ U q ( gl n ) is uniquely defined bythe following conditions: (1) There is a highest weight vector m λ ∈ M λ,t satisfying: e i · m λ = 0 , (1 ≤ i ≤ n − q ǫ i · m λ = t ρ i · q · ( λ i − ρ i ) · m λ , (1 ≤ i ≤ n ) (2) g g · m λ is a bijection from C ( t ) ⊗ U − q ( gl n ) to M λ,t , where U − q ( gl n ) denotes thesubalgebra generated by f , . . . , f n − . Under the identification t = q k , where k is now a formal parameter, M λ,t is isomorphicto the Verma module of weight λ + ( k − ρ (see e.g. [2]). Let us attempt to clarify therelationship between M λ,t and the finite-dimensional modules V λ +( k − ρ .Firstly, for k ∈ N there is a quotient mapping C ( t ) ⊗ U q ( gl n ) → U q ( gl n ) sending t → q k .Moreover, for k ∈ N then there is a C ( q )-linear map α k : M λ,t → V λ +( k − ρ which iscompatible with the module structures in the sense that the following diagram commutes: (cid:0) C ( t ) ⊗ U q ( gl n ) (cid:1) ⊗ M λ,t ( t q k ) ⊗ α k (cid:15) (cid:15) g ⊗ v g · v / / M λ,tα k (cid:15) (cid:15) U q ( gl n ) ⊗ V λ +( k − ρ g ⊗ v g · v / / V λ +( k − ρ (We take the convention here that V λ +( k − ρ = { } if λ + ( k − ρ / ∈ P + , which can occur foronly finitely many k ). The kernels of α k form a decreasing sequence of subspaces of M λ,t ,and \ k ≥ ker α k = { } . The existence of α k satisfying the above conditions determines the module M λ,t uniquely.The analogue of the finite-dimensional module U ≃ V ( k − · ( n − , ,..., is as follows: Definition 5.2.
The module W t over C ( t ) ⊗ U q ( gl n ) is the degree zero subspace of Laurentpolynomials W t = { p ( x ) ∈ C ( t ) ⊗ C ( q )[ x ± x ± · · · x ± n ] | deg p = 0 } , with the following action of the generators of U q ( gl n ) : ǫ i ( p ( x )) = x i · ∂∂x i p ( x ) e i ( p ( x )) = x i x i +1 · ( tq − ) p ( x , . . . , qx i +1 , . . . , x n ) − ( t − q ) p ( x , . . . , q − x i +1 , . . . , x n )( q − q − ) f i ( p ( x )) = x i +1 x i · ( tq − ) p ( x , . . . , qx i , . . . , x n ) − ( t − q ) p ( x , . . . , q − x i , . . . , x n )( q − q − )This is isomorphic to the module denoted W k in [2], and is irreducible over C ( t ) ⊗ U q ( gl n ).If k ∈ N is a fixed integer, we can quotient W t by the relation t = q k to obtain an infinite-dimensional module over U q ( gl n ). This is no longer irreducible, and the subspace spannedby p ( x ) with ( x . . . x n ) ( k − · p ( x ) ∈ C ( q )[ x , . . . , x n ] is identified with the module U .It is shown in [2] that there is a unique intertwining operator e φ : M λ,t → M λ,t ⊗ W t if andonly if λ ∈ P + . Moreover, for k ∈ N the intertwining operators e φ and φ ( k ) are compatiblein the sense that the following diagram commutes:(10) M λ,t e φ / / α k (cid:15) (cid:15) M λ,t ⊗ W tα k ⊗ (Proj U ◦ ( t q k )) (cid:15) (cid:15) V λ +( k − ρ φ ( k ) / / V λ +( k − ρ ⊗ U The weight-zero subspace of W t is one dimensional, which allows us to define the tracefunction e Φ( x ; q, t ) ∈ C ( q, t )[[ x , . . . , x n ]] of e φ . The compatibility (10) implies the relation e Φ( x ; q, q k ) = Φ ( k ) ( x ) mentioned in Theorem 1.1.The analogue of the Gelfand-Tsetlin basis for M λ,t is obtained by iterating the multiplicityone decomposition over U ( gl n − ), as in the finite-dimensional case. N THE EXPANSION OF CERTAIN VECTOR-VALUED CHARACTERS OF U q ( gl n ) 15 Proposition 5.3.
We have the following restriction rule for M λ,t as a module over C ( t ) ⊗ U q ( gl n − ) ⊂ C ( t ) ⊗ U q ( gl n ) : (cid:0) C ( t ) ⊗ U q ( gl n − ) (cid:1) M λ,t ≃ M µ ∈P ( n − µ ⊂ λ M µ − ( ) ( n − ,t By iterating the restriction rule above we obtain a basis for M λ,t which is indexed by chains λ = µ (0) ⊃ µ (1) ⊃ · · · ⊃ µ ( n − with µ ( i ) ∈ P ( n − i ) . We refer to this as the Gelfand-Tsetlinbasis for M λ,t .Proof. This is well known to experts. It can be proved using the maps α k and the decom-position of V λ +( k − ρ over U q ( gl n − ). (cid:3) Theorem 5.4.
With respect to the Gelfand-Tsetlin basis of M λ,t , the diagonal coefficientof the intertwining operator e φ corresponding the the chain λ = µ (0) ⊃ µ (1) ⊃ · · · ⊃ µ ( n − is equal to: Y ≤ i ≤ n c µ ( i − ,µ ( i ) ( q, t ) , µ ( i ) ∈ P + for ≤ i ≤ n − , otherwise . Proof.
Using the compatibility (10), and Theorem 1.3, one can see that the formula holdsfor t = q k when k ∈ N is sufficiently large. Since the coefficient is a rational function of q, t this determines it uniquely, and the result follows. (cid:3) References [1] V. G. Drinfeld,
Quantum groups , Proc. Int. Congr. Math., Berkely, 1986, pp 798-820.[2] P. I. Etingof and A. A. Kirillov, Jr.,
Macdonald’s polynomials and representations of quantum groups ,Math. Res. Let. (1994), 279-294.[3] M. A. Jimbo, A q-difference analogue of Ug and the Yang-Baxter equation , Lett. Math. Phys. (1985),62-69.[4] M. Kashiwara, On crystal bases , In
Representations of groups (Banff, AB, 1994) , volume 16 of CMSConf. Proc., pages 155-197. Amer. Math. Soc., Providence, RI, 1995.[5] A. N. Kirillov,
New combinatorial formula for modified Hall-Littlewood polynomials, in q-Series from aContemporary Perspective, pp. 283-333, Contemp. Math. Vol. 254, AMS, Providence, RI, 2000.[6] T. H. Koornwinder, Askey-Wilson polynomials for root systems of type BC , in Hypergeometric functionson domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), vol. 138 of Contemp.Math., Amer. Math. Soc., Providence, RI, 1992, 189–204.[7] M. Konvalinka and A. Lauve, Skew Pieri rules for Hall-Littlewood functions , DMTCS proc. AR , 2012,459-470.[8] A. Lascoux and S. O. Warnaar, Branching rules for symmetric Macdonald polynomials and sl n basichypergeometric series , Advances in Applied Mathematics (2011), 424-456.[9] I. G. Macdonald, Spherical functions on a group of p -adic type , Ramanujan Institute, Centre for Ad-vanced Study in Mathematics,University of Madras, Madras, 1971. Publications of the RamanujanInstitute, No. 2.[10] I. G. Macdonald, Symmetric functions and Hall polynomials , Oxford Mathematical Monographs, Ox-ford University Press, New York, second ed., 1995.[11] E. M. Rains, BC n -symmetric polynomials , Transform. Groups (2005), 63-132.[12] S. O. Warnaar, Remarks on the paper “Skew Pieri rules for Hall-Littlewood functions” by Konvalinkaand Lauve , Journal of Algebraic Combinatorics (2013), 519-526. Department of Mathematics, MIT, Cambridge, MA 02139
E-mail address ::