A Generalization of Tokuyama's Formula to the Hall-Littlewood Polynomials

A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's parametrization for the Schur polynomial. We generalize Tokuyama's formula to the Hall-Littlewood polynomials by extending Tokuyama's statistics. Our result, in addition to specializing to Tokuyama's result and the aforementioned classical results, also yields connections to the monomial symmetric function and a new deformation of Stanley's formula.


Introduction
Symmetric polynomials have proven an invaluable tool in the study of group representations. A particularly important class is the Schur polynomials, a ring of polynomials indexed by partitions which serve as a Z-basis for the graded ring of symmetric functions Λ [Mac88]. Schur functions, computed via the Weyl character formula, encode the characters of irreducible representations of general linear groups. Tokuyama gave a deformation of the Weyl character formula for GL(n, C). His formula expresses Schur polynomials in terms of statistics obtained from strict Gelfand-Tsetlin (GT) patterns and includes several other classical results as specializations. One is the Gelfand parametrization of the weight vectors of general linear groups (see [GZ50]); the other is Stanley's formula for the Schur q-functions (see [Sta86], [Tok88] for more detail).
Though Tokuyama's formula has been extended to other types of groups, e.g. in [HK02], analogous results for other classes of symmetric functions remain comparatively unexplored. The Hall-Littlewood polynomials are another class of symmetric functions which generalize the Schur polynomials through a deformation along a parameter t [Mac79, Chapter III], making them natural to consider in generalizing Tokuyama's formula. As well as forming a Z[t]-basis of Λ which interpolates between the dual bases of the Schur functions and the monomial symmetric functions at t = 0 and t = 1 respectively, these polynomials are used to determine characters of Chevalley groups over local and finite fields [Tok88]. Stanley's formula, a specialization of Tokuyama's, expresses the Hall-Littlewood polynomials at the singular value t = −1 (commonly known as the Schur q-functions, [Mac79, Chapter III]) as a summation over strict GT patterns. However, in existing literature there does not exist an analogue of Tokuyama's formula capable of expressing the general Hall-Littlewood polynomials as a summation over combinatorial data from Gelfand-Tsetlin patterns.
In this paper, we provide such a deformation. This formula, in addition to linking the classical specializations of Tokuyama, also includes the formulas for Schur q-functions and the monomial symmetric functions as specializations.
Section 2 introduces the Schur and Hall-Littlewood polynomials as well as Tokuyama's result. Section 3 defines a set of raising operators and presents a related proposition. In Section 4 we extend Tokuyama's combinatorial notations on Gelfand Tsetlin patterns and state the main theorem of this paper; Section 5 presents the proof of this theorem. We also include a computed example of a case of this theorem for clarity in the Appendix. Section 6 provides an alternate statement of this main result. We conclude in Section 7 with interesting consequences and specializations of the results.
The authors would like to thank Professor Paul Gunnells for suggesting this area of research and for his invaluable guidance; and are likewise grateful to Lucia Mocz for her dedicated mentoring. Finally, the authors would like to thank Professor Glenn Stevens and the PROMYS program, without which this research could not have been pursued.

Preliminary Notation and a Theorem due to Tokuyama
A partition λ = (λ 1 , λ 2 , . . . , λ n ) is a finite tuple of nonnegative integers. Unless otherwise stated, a partition will be assumed to be weakly decreasing with λ i ≥ λ i+1 for all i. The length of a partition is the number of entries in the tuple and the size is |λ| = n i=1 λ i . Partitions of equal length are added component-wise, and given two partitions λ and µ of lengths n and m respectively, we express their concatenation as λ µ = (λ 1 , . . . , λ n , µ 1 , . . . , µ m ). A partition of particular importance is (2.1) ρ n = (n − 1, n − 2, . . . , 1, 0).
We often drop the subscript n of ρ n when its value is clear from context. Let the polynomial f (x) be short for f (x 1 , . . . , x n ) and x λ = x λ 1 1 x λ 2 2 . . . x λn n . An element σ of the symmetric group S n acts on a polynomial by permuting the variables x i and on a partition by permuting the parts λ i . We use sgn σ as the standard sign function. We denote the Weyl denominator as ∆ n given by and define a deformation ∆ n (t) by Note that ∆ n (1) = ∆ n and ∆ n (0) = x ρ .
Theorem 2.1. By the Weyl character formula, the Schur polynomial corresponding to the partition λ of length n is We define the Hall-Littlewood polynomials below, noting that it differs from the original definition given in [Mac79] by the omission of a stabilizing factor.
Definition 1. The Hall-Littlewood polynomial for a partition λ of length n is Note: The traditional Hall-Littlewood polynomials defined in [Mac79] are for some suitable stabilizing factor v λ (t). Since the stabilizing factor may simply be adjoined to our results if necessary, we choose to omit it for the purpose of this paper, and refer to the polynomials R λ (x; t) as the Hall-Littlewood polynomials. It is not difficult to see that R λ (x; 0) = s λ (x) and R λ (x; 1) = σ∈S k σ(x λ ), the latter being the monomial symmetric function m λ (x) without its stabilizer.
Definition 2. A Gelfand-Tsetlin (GT) pattern is a triangular array of nonnegative integers of the form: a 1,1 a 1,2 a 1,3 . . . a 1,n a 2,2 a 2,3 . . . a 2,n . . . . . . . . . a n−1,n−1 a n−1,n a n,n where each row is weakly decreasing and each a i,j for i > 1 satisfies We take G(λ) to be the set of all GT patterns of top row λ. A strict GT pattern is one in which each row is strictly decreasing; we take SG(λ) ⊆ G(λ) to be the set of all strict GT patterns with top row λ.
Furthermore, for some GT pattern, we define the tuple which can be viewed as the ith row of the pattern expressed as a tuple. We introduce Tokuyama's formula by presenting several statistics on entries in a GT pattern used in [Tok88].
The quantities l(T ), r(T ) and s(T ) denote the numbers of left-leaning, right-leaning and special entries in a GT pattern respectively.
For a GT pattern with n rows, the statistic m i (T ) is defined as and consequently the n-tuple m(T ) is taken to be (2.10) m(T ) = (m 1 (T ), . . . , m n (T )) .
We are now able to state a theorem due to Tokuyama [Tok88], which expresses the Schur polynomials in terms of statistics from Gelfand-Tsetlin Patterns: Tok88]). For any decreasing partition λ of length n, we have

Raising Operators
In this section we present and prove an important proposition required for the proof of this paper's results. This proposition expresses certain Hall-Littlewood polynomials corresponding to increasing partitions as Hall-Littlewood polynomials of weakly decreasing partitions by using raising operators [Mac79, Chapter I].
The length of a raising operator φ, denoted l(φ), is defined as the number of operators in the minimal decomposition of φ into elementary operators of the form [i i + 1]. The identity raising operator Id is assigned length zero.
Definition 5. Given a partition λ of length n, we recursively define Ω(λ) to be the set of raising operators such that • The identity raising operator Id ∈ Ω(λ), and • for all raising operators φ ∈ Ω(λ), if the partition φ(λ) With these notions, we demonstrate an interesting quality of Ω(λ) used in connection with the Hall-Littlewood polynomials which proves useful in Theorem 4.1.
Proposition 3.1. For all φ ∈ Ω(λ + ρ), we have Proof. Let α = λ + ρ. It is clear that 3.2 holds for φ = Id. Therefore, due to the recursive definition of Ω(α), it suffices to prove that for any elementary operator [i i + 1] acting upon consecutive parts α i and α i+1 in the tuple α such that α i = α i+1 + 2, or equivalently We now prove (3.3). Let i+2 · · · x λn n ), noting that both g(x) and f (x) are unchanged by the permutation (i i + 1). Then, setting λ i = a + 1 and λ i+1 = a, we see that We know that if for some polynomial h(x) there exists a transposition (i j) ∈ S n such that (i j)(h(x)) = h(x), then the identity σ∈Sn (sgn σ) σ (h(x)) = 0 holds. Therefore, since tf (x)g(x)x a+1 i x a+1 i+1 is unchanged by the permutation (i i + 1), we have Similarly, we can find that Replacing the permutations σ with σ(i i + 1) in the right hand side of (3.5) returns the right hand side of (3.4) multiplied by t. Thus, substituting this with the left hand side of (3.4) returns Dividing through by ∆ n gives us the desired result.

Statistics on Gelfand-Tsetlin patterns and Main Theorem
To state the results of this paper, we require some notation. Instead of labelling each entry as left-leaning, right-leaning or special as in Definition 3, we label each entry with both a left-sided property and a right-sided property. The left-sided property encodes the relationship the entry a i,j has to the entry directly above it and to its left, namely a i−1,j−1 . Similarly, the right-sided property encodes the relationship that a i,j has to a i−1,j . The left-sided properties of a entry p l (a i,j ) are assigned as and similarly, the right-sided properties of an entry p r (a i,j ) are assigned as We may then define two functions using these properties.
Definition 6. The chain of an entry a i,j with i > 1 is With these we define two more functions: the first is a generalization of the expressions (−q) and (1 − q) from Tokuyama's formula; and the second considers two entries of a GT pattern that are related to a i,j , in particular a i+1,j and a i+1,j+1 .
Definition 7. For an entry a i,j with i > 1, we define For an entry a i,j with i < j < n, we define Example 4.1. Given the following segment of a GT pattern: 5 3 1 4 3 We see that the 4 has properties al and ar. Thus For the entries in the second row, we find g(p r (4)) = −qt and g(p l (3)) = −q, thus the 3 in the first row gives d(3) = (−qt) · (−q) = q 2 t. Table 1 presents all possible values for w(a i,j ) and d(a i,j ) that we may need to consider. Table 1. List of possible w(a i,j ) and d(a i,j ) values for an entry a i,j .
Example 4.1 illustrates that to define w(a i,j ) and d(a i,j ), we only need to know two consecutive rows of a GT pattern. This leads us to the following definition.
Definition 8. Suppose λ is a strictly decreasing partition. We define G 2 (λ) to be the set of all partitions µ such that the length of µ is one less than that of λ, and λ i ≥ µ i ≥ λ i+1 for all i. For λ of length 1, we let G 2 (λ) = {∅}.
This definition ensures that µ would be a valid weakly decreasing row directly below a row λ in a GT pattern. Arranging λ and µ in this manner, we are able to extend Definition 7 to parts in λ and µ, and thus define w(µ i ) for all i and d(λ i ) for 1 < i < n. We require these in a special matrix.
Thus, with these definitions, we are able to state a theorem which generalizes Theorem 2.2 due to Tokuyama to the Hall-Littlewood polynomials.
Theorem 4.1. For a partition λ of length n, we have Note: We include an example of a computation of Theorem 4.1 for a particular GT pattern in Appenix A. Readers may prefer to visit this before continuing to view the theorem's proof in the next section.

Proof
In this section, we prove Theorem 4.1. We begin by providing the reader with an outline of the proof: GT patterns are inductive in nature because removing a top row reveals a smaller nested GT pattern. This yields a recursive statement of Theorem 4.1 which we present and prove as Theorem 5.1. To prove this, we further notice that cofactor expansion upon the matrix in Definition 9 yields another recursive aspect of the Hall-Littlewood polynomials, which links the Hall-Littlewood polynomials corresponding to the partition λ to those of the same partition with the first one or two parts removed. This result is given algebraically in Lemma 5.1 and Corollary 5.1 to follow. We use these ideas to show our main result.
Theorem 4.1 will be proven as a corollary to our next theorem, which is expressed as a summation over the set G 2 (λ + ρ). This theorem is based on the idea that GT patterns contain smaller nested GT patterns, which all have top rows that are elements of G 2 (λ + ρ).
Prior to its statement we define some operators.
Theorem 5.1. For a partition λ of length n, we have We will prove Theorem 5.1 by induction. It is easy to check that the base cases of n = 1 and n = 2 hold by considering all the possible sets of G 2 (λ + ρ) corresponding to λ with length 1 or 2 and their associated coefficients. We present the inductive hypotheses and several relevant lemmas before proving the inductive step to conclude our result.
Our inductive hypotheses for n > 2 are Assuming these, we prove the formula in Theorem 5.1 for n > 2.
Lemma 5.1. R λ (x; t) can be expressed in terms of λ 1 and R λ 1 (x; t) as Proof. Let the permutation ψ i = (i i − 1 · · · 1), and let H be the symmetric group acting on the (n − 1) indices (2, 3, . . . , n). Then by definition Because σ ∈ H does not permute x 1 , we have which rearranges to the desired result.
Corollary 5.1. R λ (x; t) can be expressed in terms of λ 1 , λ 2 and R λ 1, 2 (x; t) as Proof. Taking an equivalent expression of Lemma 5.1 in which we express R λ 1 (x; t) in terms of λ 2 and R λ 1, 2 (x; t), we replace this into the original expression of Lemma 5.1. Rearranging gives us the desired result.
We introduce a function that will be used in several of the upcoming lemmas. It is Assuming the inductive hypotheses given in (5.5) and (5.6), we will use this notation in the following two lemmas to show a primary correspondence between the Hall-Littlewood polynomials and the set G 2 of various partitions.
Lemma 5.2. Suppose u is some nonnegative integer. Then, assuming the inductive hypothesis in (5.5), we have Proof. By applying Lemma 5.1 to write R u (µ−ρ) (x; t) in terms of u and R µ−ρ (x; t), the left hand side of the above equality becomes µ∈G 2 (λ 1 +ρ) We rearrange this expression to write this as Finally, we replace the expression to the right of ζ i using the inductive hypothesis in (5.5) to give the desired result.
Lemma 5.3. Suppose u and v are some nonnegative integers. Then, assuming the inductive hypothesis in (5.6), we have Proof. The proof of this lemma is very similar to that of the previous Lemma 5.2. For brevity we only present the outline of the proof.
Recalling the definition of F λ (κ), we rewrite the left hand side using Corollary 5.1 to express R ((u,v) (µ−ρ)) (x; t) in terms of u, v, and R µ−ρ (x; t). Then, upon rearranging (in a similar fashion to Lemma 5.2), we directly apply the inductive hypothesis in 5.6 to obtain the desired result.
The next two lemmas address each of the two determinants on the right hand side respectively.
Recalling from Definition 6 that d(λ 2 ) = g(p r (µ 1 )) · g(p l (µ 2 )), we notice that each of the four coefficients in the previous expression corresponds exactly to each of the four possible nonzero values for d(λ 2 ). Thus, we have With these lemmas in mind, we return to the proof of Theorem 5.1.
Proof. We begin with the following claim: This can easily be seen by swapping the subscripts i and j in the right hand side, revealing RHS = −RHS.
We divide through the equality above by ∆ n , altering the products and the bounds of the summation, and multiply by x 1 (1 − t) to find 0 = 2≤i≤n 2≤j≤n j =i Using the identity we break the double summation into two parts; in particular, if we take where h i,j (λ) is defined in Lemma 5.3, then Consider the claim: a =i This holds by expressing R λ 1 (x; t) of the left hand side explicitly using Lemma 5.1 and rearranging the consequent result. We notice that the right hand side of (5.19) is equivalent to setting j = 1 in the double summation of (5.18). Thus, adding either side of (5.19) to either side of (5.18) respectively, we have Multiplying through by −x λ 1 −λ 2 1 , and adding to both sides, we may apply Lemma 5.1 once on either side of the equality to obtain Recalling O i from Lemma 5.4 as we combine the two summations on the right hand side and recall L from (5.17) to write Multiplying throughout by n i=2 (x 1 − qx i ), we apply Lemmas 5.4 and 5.5 to the respective summations on the right hand side to find Finally, recalling from (5.13) that |M(λ + ρ; µ)| = w(µ 1 )|M(λ 1 + ρ; µ 1 )| − d(λ 2 )|M(λ 1, 2 + ρ; µ 1, 2 )|, we multiply by ζ(∆ n−1 ) to conclude ∆ n (q) · R λ (x; t) = µ∈G 2 (λ+ρ) |M(λ + ρ; µ)|x |λ+ρ|−|µ| 1 ζ(∆ n−1 (q) · R µ−ρ (x; t)).
Theorem 4.1. For a partition λ of length n, we have Proof. We prove this using mathematical induction, where our inductive hypothesis is that Theorem 4.1 holds for all Hall-Littlewood polynomials in (n − 1) variables. The base case formula of one variable is easy to check. We now prove the formula holds for n > 1 assuming the inductive hypothesis. We notice that all elements µ ∈ G 2 (λ + ρ) in Theorem 5.1 have the property that λ + ρ is strictly decreasing, whereas µ need not be. If µ is not strictly decreasing, the computed Hall-Littlewood polynomial R µ−ρ (x; t) has an increasing partition µ − ρ. Proposition 3.1 allows us to express these Hall-Littlewood polynomials in terms of some Hall-Littlewood polynomials that are obtained from strictly decreasing κ ∈ G 2 (λ + ρ). In particular we have that for all non-strict µ ∈ G 2 (λ + ρ), there exists a unique strict κ ∈ G 2 (λ + ρ) and a unique element φ ∈ Ω(κ) such that φ(κ) = µ. Furthermore any tuple φ(κ) obtained from φ ∈ Ω(κ) will either be a valid element of G 2 (λ + ρ) or will cause |M(λ + ρ; φ(κ))| = 0 and can be neglected. Hence, we may rewrite Theorem 5.1 as the following summation over strictly decreasing µ ∈ G 2 (λ + ρ): By applying the inductive hypothesis to the Hall-Littlewood polynomials R µ−ρ (x; t) of (n−1) variables in the identity above, we are reduced to the formula in Theorem 4.1.

Complements
Examining Theorem 4.1, we see that for each given GT pattern, we compute a product ranging from i = 1 to i = n − 1, which is essentially an iteration through each row. However, in Theorem 2.2, Tokuyama does not need this iteration and is able to consider each GT pattern as a whole. Thus, we provide an alternate statement of Theorem 4.1 that removes the need to consider each row individually.
Instead of considering the set Ω of each individual row, we extend its definition to an entire strict GT pattern T by taking the Cartesian product Let the (n − 1)-tuple φ T be an element of Ω(T ), such that φ T = (φ 2 , φ 3 , . . . , φ n ). Then φ T acts on T canonically by having every φ i in the tuple act on the row r i . We see that the action of φ T on a GT pattern T reveals a new pattern, which we denote as φ T (T ). Figure 1 provides an example of a GT pattern and a possible corresponding pattern φ T (T ). It is worth noticing that the pattern R T (T ) is not necessarily a GT pattern as it need not satisfy the condition a i−1,j−1 ≥ a i,j ≥ a i−1,j . The matrix M(r i ; φ(r i+1 )) in Theorem 4.1 implicitly considers the relationship between entries in row r i+1 after the action of the raising operator φ, and entries in the row r i in its original form. When we extend the definition of the set Ω to act on the entire pattern T , therefore acting on both rows r i and r i+1 at once, we modify the definitions of the left-sided property, right-sided property, chain and slice to account for this change.
Definition 10. For an entry a i,j in T , let a ′ i,j denote the corresponding entry in φ T (T ) after the action of the raising operator φ T on T . Now, we reassign the left-sided properties of an entry p l (a ′ i,j ) as follows: Similarly, we reassign the right-sided properties of an entry p r (a ′ i,j ) as follows: Note that these are the same definitions given in (4.1) and (4.2), with the only difference being that we compare the entry a ′ i,j of a pattern after it has been acted on by the raising operator φ T , to the appropriate entry in the row r i−1 of the original GT pattern.
In a similar manner to the above, we take the chain to be a function of a ′ i,j , with the only alteration from Definition 6 being that the chain of any entry with an outward property is zero i.e. c(o) = 0. We also extend slice such that g(o) = 0.
We observe that the outward label only serves to implicitly disallow any φ T (T ) patterns which contain a row φ i (r i ) / ∈ G 2 (r i−1 ) by reducing its coefficient to zero. This is analogous to defining M(λ; µ) to be the (n − 1) by (n − 1) zero matrix for any µ / ∈ G 2 (λ). Let S denote a tuple of entries a ′ i,j with i > 1 in some pattern φ T (T ). The ordering of S is specified as follows: if a ′ i 1 ,j 1 precedes a ′ i 2 ,j 2 in S, then either i 1 < i 2 , or both i 1 = i 2 and j 1 < j 2 . We also have the complementary tuple of S, denoted as S, that contains all entries a ′ i,j with i > 1 that are not in S. Then, given a GT pattern T , we define S(T ) to be the set of all tuples S of T , and we similarly define S(φ T (T )).
Let Z(S) be the set of all tuples obtained by the following Cartesian product maintaining the order of S: Given a tuple z ∈ Z(S), we define χ(z) as (6.5) χ(z) = 0 if there exists p r (a ′ i,j ) and p l (a ′ i,j+1 ) in z 1 otherwise .
In this manner, every element of Z(S) can be viewed as a choice of either the left-sided or the right-sided property of each entry in S, with χ(z) acting as a restriction on the possible choices. With these in hand, we have the following definition: Definition 11. For z ∈ Z(S) we define G(S) as where G(S) = 1 when S = ∅.
Example 6.2. Using the GT pattern in Example 6.1, take S = (5, 1). We then see that Additionally we have the following definition: Definition 12. Given a tuple of entries S from a GT pattern, we have where C(S) = 1 when S = ∅.
With these definitions in hand, we are able to state a complement to Theorem 4.1.
Theorem 6.1. For a partition λ of length n, we have The new coefficient obtained from the summation of C(S)G( S) can be viewed as an expansion of the determinants of M(r i ; φ(r i+1 )) in Theorem 4.1. In this manner, Theorem 6.1 provides more insight into the structure of the matrix M(r i , φ(r i+1 )) by elucidating the choosing process that occurs during the computation of the determinant of M. This theorem also eliminates the need to consider each row separately by extending the set Ω to act on the entire GT pattern, as opposed to one row at a time. It thus offers an interesting perspective on Theorem 4.1.

Specializations
The results of this paper generalize Tokuyama's formula and several other existing results.
We demonstrate a few of these specializations.
Tokuyama's formula. We know that for all raising operators φ = Id, the length of φ is at least 1. Therefore, examining Theorem 4.1, setting t = 0 reduces the equation to As these are all the identity cases on strict Gelfand-Tsetlin patterns, every row is strictly decreasing. This implies that consecutive entries cannot have p r (a i,j ) = r and p l (a i,j+1 ) = l, and consequently d(a i,j ) cannot be −q. All the remaining possibilities of d(a i,j ) are reduced to 0 when t = 0. Thus, if we let r i+1 = (µ 1 , . . . , µ n−i ), the matrix M(r i ; r i+1 ) simplifies to Therefore, we have Finally, returning to Tokuyama's terminology of left-leaning, right-leaning and special entries from Definition 3, we find that w(a i,j ) simplifies to and, substituting this into (7.1) from earlier, we conclude with Tokuyama's formula: Comparing the results of this paper with Tokuyama's formula reveals some interesting distinctions regarding the structure of the Hall-Littlewood polynomials in relation to the Schur polynomials. Theorem 5.1 demonstrates that when expressing R λ (x; t) recursively in terms of R µ−ρ (x; t), it is more natural to include several non-strictly decreasing partitions µ in the summation. This is not an issue for Tokuyama's formula as Schur polynomials of such non-strictly decreasing partitions µ are just s µ−ρ (x) = 0.
In fact, whilst Theorems 4.1 and 6.1 are stated as summations over strict GT patterns, the use of Ω is to allow an implicit summation over all possible non-strict rows. We thus naturally seek a way to consider the contributions of such rows directly, eliminating the more ad hoc use of Ω. Both theorems also highlight the added complexity in a Hall-Littlewood polynomial as they account for the ordering among the entries in a GT pattern instead of simply counting entries as Tokuyama does with s(T ) and l(T ).
Stanley's formula. In [Sta86], Stanley gave a formula for the Hall-Littlewood polynomials at the singular value t = −1, also known as the Schur q-functions, as a generating function of strict GT patterns of top row λ: Tokuyama subsequently showed in [Tok88] that his formula yields (7.3) when the deformation parameter q is set to −1. Theorems 4.1 and 6.1 thus specialize to (7.3) at t = 0 and q = −1, by virtue of specializing to Tokuyama's result. However, setting t to −1 in Theorem 4.1 also gives a deformation along q of (7.3), and we can show that this deformation reduces to (7.3) at q = 0. Examining Theorem 6.1, we see that any pattern containing an entry with p l = l gives an overall coefficient of zero, so we may simply sum over the set SGl(λ + ρ) ⊂ SG(λ + ρ) that contains all GT patterns without l entries. Also, since for every non-trivial φ ∈ Ω(T ) the pattern φ(T ) contains either an l or an o entry, we need only consider φ = Id. Furthermore, Theorem 4.1 reveals that any GT pattern containing consecutive r then al entries yields a coefficient D = n i=2 |M(r i−1 ; r i )| = 0 as follows: Assume for the sake of contradiction that there is a GT pattern with consecutive r then al entries for which D = 0. Let a i,j and a i,j+1 be the lowest consecutive r then al entries (there may be others on the same row, but none below). Then the r entry a i,j = u for some u ∈ N, and the al entry a i,j+1 = u − 1. Now consider the entry a i+1,j+1 . Since it cannot be l, or else D = 0, we must have a i+1,j+1 = u − 1. Then w(a i+1,j ) = 0; so to ensure M(r i ; r i+1 ) = 0 (and consequently that D = 0), we must have that either p r (a i+1,j ) = r or p l (a i+1,j+2 ) = al. But this contradicts our hypothesis. Therefore, any GT pattern containing consecutive r then al entries yields an overall coefficient of zero.
Let the set SGl * (λ + ρ) ⊂ SGl(λ + ρ) contain all GT patterns without l entries or consecutive r then al entries. Then Theorem 4.1 simplifies to where the product is taken over all possible a i,j . This leads us to introduce a bijective mapping, similar to one used in [Tok88]: After applying this mapping, we have that w(a i,j ) = 2 for all special entries and w(a i,j ) = 1 otherwise, and thus (7.4) reduces to x ρ · R λ (x; −1) = x ρ · T ∈SG(λ) 2 s(T ) x m(T ) , Dividing out by x ρ , we obtain (7.3).
Monomial symmetric function. Theorem 6.1 also simplifies significantly in the case of t = 1. Recalling Definition 6 of c(a i,j ), we see that setting t = 1 results in c(a i,j ) = 0 for any entry a i,j . Thus, the function C(S) from Definition 12 is only nonzero when S = ∅. Then, we have where K is the tuple containing all entries of φ T (T ). Therefore, we can write (7.6) ∆ n (q) · m λ = T ∈SG(λ+ρ) x m(T ) φ T ∈Ω(T )

G(K).
We also observe that G(K) is a polynomial in q that can be written as some summation of the form G(K) = (−q) k . Thus specializing q, particularly to q = 0 or q = −1, will reduce this expression further. (1 − q) 2 (1 + t)x 2 1 x 2 x 3 , to the summation. As this is the unique GT pattern with top row λ + ρ = (2, 1, 0) and m(T ) = (2, 1, 1), we should find that (A.5) gives the coefficient of x 2 1 x 2 x 3 in the expansion of ∆ 3 (q) · R (1,0,0) (x; t). The reader can verify this is indeed the case.