Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials

We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials P_lambda/mu(x;t) and Hivert's quasisymmetric Hall-Littlewood polynomials G_gamma(x;t). More specifically, we provide: 1) the G-expansions of the Hall-Littlewood polynomials P_lambda, the monomial quasisymmetric polynomials M_alpha, the quasisymmetric Schur polynomials S_alpha, and the peak quasisymmetric functions K_alpha; 2) an expansion of P_lambda/mu in terms of the F_alpha's. The F-expansion of P_lambda/mu is facilitated by introducing starred tableaux.

the Schur functions in terms of fundamental quasisymmetric functions F α . This expansion, which follows from the technique of standardization, is indicated by the vertical line connecting s µ and F α in Figure 1. Such standardizations have been used recently to give F -expansions of various symmetric functions including plethysms of Schur functions [14], the modified Macdonald polynomials [4,5], the Lascoux-Leclerc-Thibon (LLT) polynomials [10], and (conjecturally) the image of a Schur function under the Bergeron-Garsia nabla operator [13].
Given Hivert's construction, the following question arises. Is there an expansion of the P λ in terms of the G γ which would interpolate between the F -expansion of the s λ at t = 0 and the M -expansion of the m µ at t = 1? The main purpose of this paper is to provide such an expansion, and also to provide other change-of-basis matrices between different bases of the Hall algebra and the algebra of quasisymmetric functions, as explained below. In terms of Figure 1, we provide the middle vertical edge as well as the two downward directed edges in the bottom face (namely, from G γ (t) to both F α and M β ).
G-expansion of the P Basis. In Theorem 5.6 we give an explicit combinatorial expansion of the Hall-Littlewood polynomials P λ (x; t) in terms of the Hivert quasisymmetric Hall-Littlewood polynomials G γ (x; t). This provides the desired t-interpolation between Gessel's F -expansion of Schur polynomials (i.e., t = 0) and the obvious expansion of m λ 's into M α 's (i.e., t = 1).
F -expansion of the P Basis. One of the main tools for our calculations is the definition of a new class of tableaux, called starred tableaux. With these, we give in Theorem 4.1 a combinatorial expansion of the skew Hall-Littlewood polynomials P λ/µ (x; t) in terms of the fundamental quasisymmetric functions F α . A minor variation to our method gives a corresponding expansion for the dual Hall-Littlewood polynomials Q λ/µ . G-expansion of the F and M Bases. In Theorems 5.1 and 5.3 we give explicit combinatorial expansions for the F α and the M β in terms of the G γ . These are inverse matrices to those found in [7].
The structure of this extended abstract is as follows. The bases discussed are defined in §2 while the known transition matrices are summarized in §3. The expansions of the Hall-Littlewood polynomials in terms of the F α and G γ (x; t) are presented in §4 and §5, respectively.
This text is an extended abstract of the preprint [12], where complete proofs can be found. Furthermore, in [12] we give explicit combinatorial expansions for the peak quasisymmetric functions K α and the quasisymmetric Schur functions S β in terms of the G γ .

Review of Symmetric and Quasisymmetric Bases
This section reviews the definitions of the symmetric and quasisymmetric functions appearing in Figure 1.
Logically, the precise definitions of the various bases are not needed in this paper, as the expansions found in §4 and §5 are derived from the known transition matrices of §3. However, the material of this section is included for completeness.

Compositions and Partitions
Given n ∈ N, a composition of n is a sequence α = (α 1 , α 2 , . . . , α k ) of positive integers (called parts) with α 1 +· · ·+α k = n. Define the length (α) to be the number of parts of α, and the size |α| to be the sum of its parts. For example, the composition α = (2, 4, 1) has (α) = 3 and |α| = 7. We may abbreviate the notation, writing α as 241, when no confusion can arise. Let Comp n be the set of compositions of n, and let Comp be the set of all compositions. A composition λ = (λ 1 , λ 2 , . . . , λ k ) ∈ Comp n is called a partition of n iff λ 1 ≥ λ 2 ≥ · · · ≥ λ k . We write Par n for the set of partitions of n and Par for the set of all partitions.

The inverse bijection sends any subset
Given α, β ∈ Comp n , we say β is finer than α, denoted β α, iff sub(α) ⊆ sub(β). Informally, β is finer than α if we can chop up some of the parts of α into smaller pieces (without reordering anything) and obtain β. For example,

Symmetric Polynomials
Let K be a field of characteristic zero, and let S N denote the symmetric group on N letters. A polynomial Write Sym N for the ring of symmetric polynomials in N variables. For each n ≥ 0, let Sym n N be the subspace of Sym N consisting of zero and the homogeneous polynomials of degree n. For N ≥ n, bases of the vector space Sym n N are naturally indexed by partitions of n. Given λ ∈ Par n of length k ≤ N , the monomial symmetric polynomial m λ (x 1 , . . . , x N ) is the sum of all distinct monomials that can be obtained by permuting subscripts in x λ1 1 x λ2 2 · · · x λ k k . For N ≥ n, {m λ (x 1 , . . . , x N ) : λ ∈ Par n } is readily seen to be a basis of Sym n N . Now suppose N ≥ n and ν ∈ Par n is a partition with distinct parts. If necessary, we append parts of size zero to the end of ν to make ν have length N . The monomial antisymmetric polynomial indexed by ν in N variables is Given λ ∈ Par n , the Schur symmetric polynomial indexed by λ in N variables is It can be shown that this rational function is both a polynomial and symmetric. Moreover {s λ : λ ∈ Par n } is a basis of Sym n N [15, §I.3, p. 40]. For the rest of the paper, let t be an indeterminate, and let K be any field containing Q(t) as a subfield. Following [15, §III.1, pp. 204-7], we define the Hall-Littlewood symmetric polynomials as follows. Fix λ ∈ Par n and N ≥ n. Extend λ to have length N by appending parts of size zero if needed. Define Given that λ has m 0 parts equal to 0, m 1 parts equal to 1, and so on, it can be shown that R λ is divisible by We then define the Hall-Littlewood polynomial It can be shown [15,§III.2,p. 209] that for N ≥ n, the set {P λ (x 1 , . . . , x N ; t) : λ ∈ Par n } is a basis for Sym n N . Moreover, setting t = 0 in P λ gives s λ , whereas setting t = 1 in P λ gives m λ . Thus, the Hall-Littlewood basis "interpolates" between the Schur basis and the monomial basis. One can define Schur polynomials and Hall-Littlewood polynomials more concretely by giving combinatorial descriptions of their expansions in terms of monomial symmetric polynomials. See §3.1 below.

Quasisymmetric Polynomials
QSym N for the ring of quasisymmetric polynomials in N variables. For each n ≥ 0, let QSym n N be the subspace of QSym N consisting of zero and the homogeneous polynomials of degree n. For N ≥ n, linear bases of QSym n N are naturally indexed by compositions of n. Symmetric polynomials are quasisymmetric, so Sym n N is a subspace of QSym n N .
where we sum over all subscript sequences w = w 1 w 2 · · · w n such that 1 ≤ w 1 ≤ w 2 ≤ · · · ≤ w n ≤ N and for all j ∈ sub(α), w j < w j+1 . In other words, strict increases in the subscripts are required in the "breaks" between parts of the composition α. A routine inclusion-exclusion argument shows that for N ≥ n, {F α (x 1 , . . . , x N ) : α ∈ Comp n } is a basis of QSym n N . Note that some authors index fundamental quasisymmetric polynomials by pairs n, T where T ⊆ [n − 1]. Additionally, various letters (F , L, Q, etc.) have been used to denote these polynomials.
As in the symmetric case, we would like to have quasisymmetric Hall-Littlewood polynomials (depending on a parameter t) that interpolate between F α (when t = 0) and M α (when t = 1). We sketch the definition of one such family of polynomials, introduced and studied by Hivert [7]. Quasisymmetric functions arise as the invariants of a certain action of S n on polynomials. From this action, one can define divided difference operators in a degenerate Hecke algebra H n (0) which can then be lifted to H n (q). Hivert's quasisymmetric Hall-Littlewood polynomials thereby arise from a corresponding t-analogue ω of the Weyl symmetrizer. For a composition α of length k ≤ N , define As in the case of symmetric Hall-Littlewood polynomials, there is a more concrete combinatorial definition of G α giving its expansion into monomials. We discuss this definition in §3.2.

Known Transition Matrices
In the theory of symmetric and quasisymmetric polynomials, much combinatorial information is encoded in the transition matrices between various bases. Given two bases B = {B λ : λ ∈ Par n } and C = {C λ : λ ∈ Par n } of Sym n N , the transition matrix M(B, C) is the unique matrix (with entries in K and rows and columns indexed by partitions of n) such that  The expansion of Schur polynomials into monomials uses semistandard tableaux. For later work, we will also need tableaux of skew shape. Suppose λ, µ ∈ Par satisfy µ ⊆ λ, i.e., µ i ≤ λ i for all i. Define the skew diagram We will draw skew diagrams using the English convention where the longest rows are at the top. For N ∈ N + , a semistandard tableau (SSYT) of shape λ/µ with entries in that is weakly increasing along rows and strictly increasing down columns. Writing n = |λ/µ|, a standard tableau (SYT) of shape λ/µ is a bijection S : λ/µ → [n] that is also a SSYT. Let SSYT N (λ/µ) be the set of all SSYT of shape λ/µ with entries in [N ], and let SYT(λ/µ) be the set of all SYT of shape λ/µ. For any T ∈ SSYT N (λ/µ), the content monomial x T is defined to be c∈λ/µ x T (c) . The skew Schur polynomial in N variables can now be defined as The ordinary Schur polynomial s λ is obtained by taking µ = (0). For λ, ν ∈ Par n and N ≥ n, it follows that M(s, m) λ,ν is the Kostka number K λ,ν , namely the number of SSYT of shape λ and content ν. Lascoux and Schützenberger [9] first discovered a combinatorial formula for the t-Kostka matrix M(s, P ) involving the famous charge statistic. Given a permutation w = w 1 w 2 · · · w n of [n], let IDes(w) be the set of k < n such that k + 1 appears to the left of k in w, and let chg(w) = k∈IDes(w) (n − k).
Next, let v be a word of partition content (i.e., for all k ≥ 1, the number of (k + 1)'s in v is no greater than the number of k's). Extract one or more permutations from v as follows. Scan v from left to right marking the first 1, then the first 2 after that, etc., returning to the beginning of v when the right end is reached. Do this until the largest symbol has been marked. Remove the marked symbols from v (in the order they appear) to get the first permutation. Continue to extract permutations in this way until all symbols of v have been used, and let chg(v) be the sum of the charges of the associated permutations. Finally, given a SSYT T of partition content, let w(T ) be the word obtained by reading symbols row by row from top to bottom, reading each row from right to left. Then define chg(T ) = chg(w(T )).
Define the weight of a special cell (i, j) to be wt((i, j)) = |{(u, j − 1) ∈ λ/µ : u ≥ i and T ((u, j − 1)) < T ((i, j))}| In other words, a cell c with entry v = T (c) is special for T iff c is not in column 1 and there are no v's in the column of T just left of c's column. In this case, the weight of c is the number of cells weakly below c in the column just left of c that either have entries less than v or are part of the diagram for µ. Now define the set of starred semistandard tableaux ). Then Macdonald's monomial expansion of the skew Hall-Littlewood polynomials is Expanding the product in ψ T (t) using the distributive law, we get E⊆Sp(T ) c∈E (−t wt(c) ). Comparing to the overall weight of starred tableaux, we find that These special cells have respective weights 1, 1, 1, 3, 2, 1, 2. So T contributes the term {(1, 4), (1,6), (2, 6)}). The overall weight of the object

M(s, F ), M(G, F ), and M(G, M )
The fundamental quasisymmetric expansion of Schur polynomials is a sum over standard tableaux, rather than semistandard tableaux. Given λ ∈ Par n and S ∈ SYT(λ), define the descent set Des(S) to be the set of k < n such that k + 1 appears in a lower row of S than k. Define the descent composition Des (S) = comp(Des(S)) to be the composition associated to this subset of [n−1]. Gessel first proved [3] that for N ≥ n = |λ|, In other words, M(s, F ) λ,α is the number of standard tableaux with shape λ and descent set sub(α). Let α, β ∈ Comp n with β finer than α. Say (α) = k and (β) = m. By definition, there exist indices records the number of parts of β derived from each part of α. Define s(α, β) = k j=1 j(i j − i j−1 − 1). Note that in the notation Bre(β, α) from [7], the finer composition is listed first, but in the function s (and g, ξ defined in §5.1), we list the finer composition second. This ordering is more convenient when working with transition matrices.

F -expansion of Skew Hall-Littlewood Polynomials
Recall from §3.1 the combinatorial formula (1) for the monomial expansion of the skew Hall-Littlewood polynomials P λ/µ (x 1 , . . . , x N ; t). This section converts this formula to an expansion of these polynomials in terms of the fundamental quasisymmetric basis. In particular, this provides a combinatorial interpretation for the entries of M(P, F ). We remark that one can also obtain M(P, F ) = M(P, s) M(s, F ) by combining Carbonara's combinatorial formula for M(P, s) in terms of special tournaments [1] with Gessel's formula (2) for M(s, F ). However, this produces a quite complicated interpretation for the coefficients in M(P, F ) as signed combinations of standard tableaux and special tournaments. The new interpretation developed below is much simpler.
To state our result, we need a few more definitions. Given a skew diagram λ/µ with n cells, let SYT * (λ/µ) be the set of starred standard tableaux S * = (S, E) such that S is a standard tableau of shape λ/µ. In this case, observe that Sp(S) consists of all cells in the diagram not in column 1. So E can be an arbitrary subset of cells of λ/µ not in column 1. Define the ascent set of S * , denoted Asc(S * ), to be the set of all k < n such that either (a) k + 1 appears in S in a lower row than k, or (b) there exist u, i, j with S((u, j − 1)) = k, S((i, j)) = k + 1, and (i, j) ∈ E. The second alternative says that k + 1 appears in a cell of E located in the next column after the column containing k. Define Asc (S * ) = comp(Asc(S * )) to be the associated composition.
We derive a similar formula for the skew Hall-Littlewood polynomials Q λ/µ in [12].
Example 4.2 Using Theorem 4.1, we can make the following calculation. Each term corresponds to the starred standard tableau shown below it: expressed the entries of the inverse t-Kostka matrix M(P, s) as signed, weighted sums of special tournament matrices. An alternative description can be obtained by following M(P, F ) by the projection from QSym to Sym given in [2]. The entry of M(P, s) λ,µ is again described as a sum of signed, weighted objects. However, in this description the objects are pairs (S * , T ) where S * ∈ SYT * (λ) and T is a "flat special rim-hook tableau" of shape µ and content Asc (S * ).
In addition to working for skew Hall-Littlewood polynomials, this new description may have computational advantages. For n = 4, there are 37 special tournament matrices that contribute to the calculation of M(P, s). However, only 23 pairs (S * , T ) are now needed. We note that these pairs do not correspond to a subclass of special tournament matrices in any simple way. Carbonara's description computes the value M(P, s) 4,22 = 0 via the fact that there are no special tournament matrices with parameters λ = (4) and µ = (2, 2). There are two such pairs (S * , T ), albeit of opposite sign and equal weight.

New Transition Matrices involving the Hivert G-basis
This section discusses combinatorial formulas for the transition matrices M(F, G), M(M, G), and M(P, G).
The idea of the proof is to show that M(G, F ) M(F, G) = I, where M(F, G) is defined above and M(G, F ) is defined in Theorem 3.3. Equivalently, we must show that for all compositions β α, is 1 for β = α and 0 otherwise. We prove this in [12] using a sign-reversing involution that cancels all negative objects.

M(M, G)
Theorem 5.3 For all α, β ∈ Comp n with β α, is We then use a counting argument to rewrite the sum as the product j:ξ α,β (j)=j (1 − t j ).
However, this can be simplified. In order to do so, we introduce some new notation. For each j ∈ sub(β), let n j = n j (β) be the number of elements of sub(β) that are at most j. Let c j = c j (S) be the unique cell of S in which j appears. Let n j = n j if j ∈ sub(β) \ Des(S) and 0 otherwise. Recall Des(S) was defined in §3.2. The idea of the proof is to group together summands in (3) indexed by the starred tableaux S * = (S, E) with the same underlying standard tableau S. A careful case analysis leads to the sum of products in (4).
Remark 5.7 If n j = wt(c j+1 ) for some j ∈ sub(β) with c j+1 ∈ Esp(S), then S can be omitted from the sum in (4).