A Quantization of a theorem of Goulden and

A theorem of Goulden and Jackson which gives interesting formulae for character immanants also implies MacMahon's Master Theorem. We quantize Goulden and Jackson's theorem to give formulae for quantum character immanants in such a way as to obtain a known quantization of MacMahon's Master Theorem due to Garoufalidis-Le-Zeilberger. In doing so, we also quantize formulae of Littlewood, Merris and Watkins concerning induced character immanants.


Introduction
Among their abundant results expressing generating functions in terms of matrix traces and determinants, Goulden and Jackson (8) obtained several identities concerning polynomials Imm λ (x) = def w∈Sn χ λ (w)x 1,w1 • • • x n,wn in x = (x 1,1 , . . ., x n,n ) whose coefficients are given by irreducible characters χ λ of S n .We will call these polynomials irreducible character immanants.Using their identities, Goulden and Jackson gave new presentations of results of Littlewood and Young and proved a generalization of MacMahon's celebrated Master Theorem.Also giving new interpretations of Littlewood's results, Merris and Watkins (13) stated similar formulae for irreducible (and other) character immanants by summing products of permanents and determinants.
Many of the above results have natural noncommutative extensions.Following authors such as Cartier-Foata (1), Garoufalidis-Lê-Zeilberger (6) and Konvalinka-Pak (10), who have stated quantum analogs of the Master Theorem, we will state quantum analogs of results of Goulden-Jackson and Merris-Watkins.As a consequence, we will also obtain a new proof of a quantized Master Theorem which is stated in (6).We review the relevant classical and quantum algebras in Sections 2-4 and state our main results in Section 5-6.
2 The symmetric group, C[x 1,1 , . . ., x n,n ], and immanant formulae Let S n be the symmetric group, and let s 1 , . . ., s n−1 be the standard adjacent tranpositions generating S n and satisfying the relations A standard action of S n on rearrangements of the word 1 • • • n is defined by letting s i swap the letters in positions i and i + 1, Thus, denoting the identity permutation of S n by e, the one-line notation of e is 12 • • • n.Using this convention, the one-line notation of vw is (vw Let (w) be the minimum length of an expression for w in terms of the generators.Equivalently, (w) is the number of inversions in the one-line notation of w.Let ≤ denote the Bruhat order on S n , i.e., v ≤ w if every reduced expression for w contains a reduced expression for v as a subword.
The ring C[x 1,1 , . . ., x n,n ] is naturally graded by degree.For each r ∈ N, the space of degree-r polynomials decomposes further into subspaces indexed by multisets of [r] = {1, . . ., r}.In particular, we will consider the immanant space span C {x 1,w1 • • • x n,wn | w ∈ S n }.For a function f : S n → C, we follow (16) in defining the f -immanant to be the element of the immanant space.Defining the notation x u,v = x u1,v1 • • • x un,vn , and x w = x e,w = x 1,w1 • • • x n,wn , we may rewrite the natural basis of the immanant space as {x w | w ∈ S n }.
Let λ = (λ 1 , . . ., λ ) be an integer partition (with > 0) and let λ denote the transpose (a.k.a conjugate) of the integer partition λ. (See (5).)Immanants Imm χ λ (x) constructed from the irreducible characters χ λ : S n → R of S n are usually abbreviated Imm λ (x), It is well known that irreducible characters are class functions on S n in the sense that if v and w have the same cycle type in S n , then χ λ (v) and χ λ (w) are equal.Equivalently, we have χ λ (v) = χ λ (w) if v = uwu −1 for some u ∈ S n .For each class function f : S n → C, we will call the immanant Imm f (x) a class immanant.Class immanants which are somewhat better understood than irreducible character immanants correspond to characters { λ | λ n} induced from the sign character of Young subgroups of S n and to characters {η λ | λ n} induced from the trivial character of Young subgroups of S n .Indeed, we have quite simple formulas for these immanants in terms of determinants and permanents of submatrices of x which we denote by x I,J = (x i,j ) i∈I,j∈J .In particular, Littlewood (11) and Merris and Watkins (13) showed that for λ = (λ 1 , . . ., λ ) n, we have where the sums are over all sequences (I 1 , . . ., I ) of pairwise disjoint subsets of Each of the sets {χ λ | λ n}, {η λ | λ n}, { λ | λ n} forms a basis for the space of class functions on S n .We may express the first basis in terms of the others by and we therefore may express irreducible character immanants in terms of the induced character immanants by The coefficients appearing in these identities are called inverse Kostka numbers and may be defined by where {ξ i | i > 0} are commuting indeterminates, and where we define ξ 0 = 1, and ξ i = 0 for i ≤ −1.

Goulden and Jackson's formulae
Goulden and Jackson identified three multivariate generating functions which are related to the irreducible character immanants {Imm λ (x) | λ n}.For the convenience of the reader, we will summarize results and proofs from (8, §2-3) in a way which will facilitate our quantum analogs in Section 5.
To begin, let t be a variable, let z = (z 1 , . . ., z n ) be a sequence of n variables, and let diag(z) be the n × n matrix whose diagonal entries are z 1 , . . ., z n .Define the sequences and by the requirement that polynomials with indices not appearing here be zero.Now define the λ 1 × λ 1 matrices A, D and the × matrices B, C by Summarizing the main results (8, Thm.2.1, Cor.2.3) on irreducible character immanants, we have the following.Given a multiset we define the K, K generalized submatrix of x to be the r × r matrix x K,K obtained from x by repeating the ith row and column k i times each.(This is called the K-replication of x in (8).)For example when K = 1112, we have k 1 = 3, k 2 = 1, and From the preceding results, Goulden and Jackson (8, Thm.3.3) derive MacMahon's Master Theorem (12) as follows.
x n,j z j kn (8) are both equal to per(x Proof: Omitted. 2

The Hecke algebra and quantum polynomial ring
In order to state and prove quantum analogs of the results of Littlewood-Merris-Watkins and Goulden-Jackson, we review some facts about the relevant algebras.
The Hecke algebra H n (q) is the C[q 1 2 , q − 1 2 ]-algebra generated either by the set The natural and modified natural generators are related by We shall call the elements {T w | w ∈ S n } and { T w | w ∈ S n } the natural basis and modified natural basis, respectively, of H n (q) as a C[q 1 2 , q − 1 2 ]-module.Specializing H n (q) at q 1 2 = 1, we obtain the classical group algebra C[S n ] of the symmetric group.
One multiplies modified natural basis elements by recursively using either of the formulas and thus obtains elements ] occurring as coeficients in the expression By the symmetry of the formulae ( 9), one sees that ]-algebra is called the quantum polynomial ring A(n; q).It is generated by n 2 variables x = (x 1,1 , . . ., x n,n ) representing matrix entries, subject to the relations for all indices i < j, k < .The quantum polynomial ring often arises in conjunction with the quantum coordinate ring of SL(n, C), which may be expressed as O q SL(n, C) ∼ = A(n; q)/(det(x; q) − 1), where det(x; q) = def w∈Sn is the quantum determinant.(We caution the reader that the second equality above is implied by the third relation in (11), and does not hold in an arbitrary noncommutative ring in n 2 variables.)Specializing A(n; q) at q ]-module, A(n; q) is spanned by monomials in lexicographic order, and we can use the relations above to convert any other monomial to this standard form.It is easy to see that the monomials Thus for all w ∈ S n we have the identity x e,w = x w −1 ,e .On the other hand, we do not in general have the equality of x v,w and x w −1 ,v −1 .A(n; q) has a natural grading by degree, A(n; q) = r≥0 A r , where A r is the C[q 1 2 , q − 1 2 ]-span of all monomials of total degree r.Furthermore, the natural basis {x a1,1 We may further decompose each homogeneous component A r by considering pairs (K, M ) of multisets of integers.Thus we obtain the multigrading where ]-span of monomials whose row indices and column indices (with multiplicity) are equal to the multisets K and M , respectively.Just as the Z-graded components A r and A s satisfy A r A s ⊂ A r+s , the multigraded components A K,M and A K ,M satisfy A K,M A K ,M ⊂ A K K ,M M , where denotes the multiset union of two multisets, ]-submodule of A(n; q) spanned by the monomials which we will again write as {x e,w = x w | w ∈ S n }.We will call elements of this submodule quantum immanants.In particular, for any C[q One important quantum immanant is the quantum determinant, which corresponds to the Hecke algebra sign character χ 1 n q : T w → (−q − 1 2 ) (w) .Another is the quantum permanent per(x; q) = w∈Sn (q 2 ) (w) x w , which corresponds to the Hecke algebra trivial character χ n q : T w → (q 2 ) (w) .

Formulae for quantum character immanants
In analogy to the S n irreducible character immanants, we will construct quantum immanants Imm χ λ q (x; q) from the irreducible characters χ λ q : H n (q) → C[q 1 2 , q − 1 2 ] of H n (q), and we will abbreviate these Two examples, as we have mentioned, are the quantum determinant and quantum permanent.Note that a Hecke algebra character χ q (i.e., the trace of any matrix representation) is not an S n -class function in the sense that χ q ( T v ) and χ q ( T u −1 vu ) are not in general equal (equivalently, χ q (T v ) and χ q (T u −1 vu ) are not in general equal).On the other hand, χ q is a class function in the sense one would expect: we have Class functions on H n (q) are completely determined by their values on natural basis elements T v for which v has the cycle notation of the form for some composition µ.This fact facilitates the creation of character tables and formulae as in ( 9), ( 14), and motivates the use of a second basis for A [n], [n] .For a permutation w with cycle notation where i j 1 is the smallest element of the j-th cycle and i [n] and reconcile the difference between S n -class functions and H n (q)-class functions as follows.
] be an H n (q)-class function.Then we have where µ(w) is the cycle type of w.
Proof: Omitted. 2 In particular, the coordinates of Imm λ (x; q) with respect to the basis {x w | w ∈ S n } are class functions on S n in the usual sense.
Quantum analogs of the immanants {Imm λ (x) | λ n} and {Imm η λ (x) | λ n} correspond to characters { λ q | λ n} and {η λ q | λ n} induced from the sign character and trivial character of Hecke algebras of Young subgroups of S n .
Let J be a subset of the standard generators {s 1 , . . ., s n−1 } of W = S n and let W J be the corresponding parabolic subgroup of W .Let W/W J be the set of cosets of the form uW J .Each such coset is an interval in the Bruhat order and thus has a unique minimal element and a unique maximal element.Let W J − be the set of minimal representatives of cosets in W/W J .It is well known that we have To prove quantum analogs of the formulae (2), we consider elements of H n (q) which are often used in conjunction with parabolic subalgebras.(See, e.g., (2), (3), (4).)For each permutation u ∈ W J − , define the Hecke algebra elements 2 ) (y) T y .
Note that if J = ∅ then each coset W/W J is a single element u ∈ S n and we have T uW J = T uW J = T u .The elements (17) are used to construct induced representations as follows.Given a partition λ = (λ 1 , . . ., λ ) of n, choose any rearrangement ν = (ν 1 , . . ., ν ) of λ and define the subset J of generators of S n by Letting H n (q) act by left multiplication on the C[q we obtain the H n (q) modules corresponding to induction of the sign and trivial characters (respectively) of the Young subalgebras of type λ.For each w ∈ S n , the two matrices representing T w have entries indexed by permutations u, v ∈ W J − , which we describe as follows.Lemma 5.2 Fix w in S n and u, v in W J − .For the above constructions of the induced sign and trivial H n (q) modules, the u, v entries of the matrices representing T w are equal to the coefficients of T w in T u T W J T v −1 and T u T W J T v −1 respectively.Proof: Omitted. 2 From this fact, we obtain the following Hecke algebra "generating functions" for induced characters.
Lemma 5.3 Let λ, ν and J be as above.Then we have w∈Sn η λ q ( T w ) T w .
Proof: Using Lemma 5.2 and summing over diagonal matrix entries, we obtain the desired equalities. 2 These Hecke algebra generating functions in turn are related to quantum immanants by the actions of H n (q) on A [n],[n] defined by x e,siv if s i v > v, x e,siv + (q x e,v • T si = x v −1 ,e • T si = x v −1 ,si = x e,vsi if vs i > v, x e,vsi + (q A straightforward but tedious computation shows that the left and right actions commute.By the definitions, it is easy to see that we have T v • x e,e = x e,e • T v = x e,v for all v ∈ S n .On the other hand, we do not in general have the equality of T v • x e,w and x e,w • T v .One consequence of the definitions (20) is the following formula.