THE QUANTUM MACMAHON MASTER THEOREM

STAVROS GAROUFALIDIS, THANG TQ LÊ, AND DORON ZEILBERGER

Abstract. We state and prove a quantum-generalization of MacMahon’s celebrated Master Theorem, and relate it to a quantum-generalization of the boson-fermion correspondence of Physics.

1. Introduction

1.1. MacMahon’s Master Theorem. In this paper we state and prove a quantumgeneralization of MacMahon’s celebrated Master Theorem, conjectured by the first two authors. Our result was motivated by quantum topology. In addition to its potential importance in knot theory and quantum topology (explained in brief in the last section), this paper answers George Andrews’s long-standing open problem [A] of finding a natural q-analog of MacMahon’s Master Theorem.

Let us recall the original form of MacMahon’s Master Theorem and some of its modern interpretations.

Consider a square matrix A = (a_{ij}) of size r with entries in some commutative ring. For 1 ≤ i ≤ r, let X_{i} := ∑_{j=1}^{r} a_{ij}x_{j}, (where x_{i}’s are commuting variables) and for any vector (m_{1}, …, m_{r}) of non-negative integers let G(m_{1}, …, m_{r}) be the coefficient of x_{1}^{m_{1}}x_{2}^{m_{r}} … x_{r}^{m_{r}} in ∏_{i=1}^{r} X_{i}^{m_{i}}. MacMahon’s Master Theorem is the following identity (see [MM]):

(1)  ∑_{m_{1},m_{2},...,m_{r}=0}^{∞} G(m_{1}, …, m_{r}) = 1/det(I − A).

There are several equivalent reformulations of MacMahon’s Master Theorem; see for example [FZ] and references therein. Let us mention one, of importance to physics.

Given a matrix A = (a_{ij}) of size r with commuting indeterminate entries which lie in a ring R, and a nonnegative integer n, we can consider its symmetric and exterior powers S^{n}(A) and Λ^{n}(A), and their traces tr S^{n}(A) and tr Λ^{n}(A) respectively. Since

tr S^{n}(A) = ∑_{m_{1}+...m_{r}=n} G(m_{1}, …, m_{r})

det(I − A) = ∑_{n=0}^{∞}(−1)^{n} tr Λ^{n}(A)t^{n},

Date: This edition: October 5, 2004. First edition: March 24, 2003.

The authors were supported in part by NSF.

1991 Mathematics Classification. Primary 57N10, 05A30. Secondary 57M25.

Key words and phrases: Boson-fermion correspondence, MacMahon’s Master Theorem, qdifference operators .

the following identity

(2)  \frac{1}{∑_{n=0}^{∞}(−1)^{n} tr Λ^{n}(A)t^{n}} = ∑_{n=0}^{∞} tr S^{n}(A)t^{n}

in R[[t]] is equivalent to (1). In Physics (2) is called the boson-fermion correspondence, where bosons (resp. fermions) are commuting (resp. skew-commuting) particles corresponding to symmetric (resp. exterior) powers.

1.2. Quantum algebra, right-quantum matrices and quantum determinants. In r-dimensional quantum algebra we have r indeterminates x_{i} ( 1 ≤ i ≤ r), satisfying the commutation relations x_{j}x_{i} = qx_{i}x_{j} for all 1 ≤ i < j ≤ r. We also consider matrices A = (a_{ij}) of r^{2} indeterminates a_{ij}, 1 ≤ i, j ≤ r, that commute with the x_{i}’s and such that for any 2 by 2 minor of (a_{ij}), consisting of rows i and i′, and columns j and j′ (where 1 ≤ i < i′ ≤ r, and 1 ≤ j < j′ ≤ r), writing a := a_{ij}, b := a_{ij′}, c := a_{i′j}, d := a_{i′j′}, we have the commutation relations:

(3)  ca = qac, (q-commutation of the entries in a column)
(4)  db = qbd, (q-commutation of the entries in a column)
(5)  ad = da + q^{−1}cb − qbc  (cross commutation relation).

We will call such matrices A right-quantum matrices.

The quantum determinant, (first introduced in [FRT]) of any (not-necessarily right-quantum) r by r matrix B = (b_{ij}) may be defined by

det_{q}(B) := ∑_{π∈S_{r}}(−q)^{−inv(π)}b_{π_{1}1}b_{π_{2}2} ⋯ b_{π_{r}r},

where the sum ranges over the set of permutations, S_{r}, of {1, …, r}, and for any of its members, π, inv(π) denotes the number of pairs 1 ≤ i < j ≤ r for which π_{i} > π_{j}.

1.3. A q-version of MacMahon’s Master Theorem. We are now ready to state our quantum version of Macmahon’s Master Theorem. Theorem 1. (Quantum MacMahon Master Theorem) Fix a right-quantum matrix A of size r. For 1 ≤ i ≤ r, let X_{i} := ∑_{j=1}^{r} a_{ij}x_{j}, and for any vector (m_{1}, …, m_{r}) of non-negative integers let G(m_{1}, …, m_{r}) be the coefficient of x_{1}^{m_{1}}x_{2}^{m_{r}} … x_{r}^{m_{r}} in ∏_{i=1}^{r} X_{i}^{m_{i}}. Let

Ferm(A) = ∑_{J⊂{1,...,r}}(−1)^{|J|}det_{q}(A_{J})

where A_{J} is the J by J submatrix of A, and

Bos(A) = ∑_{m_{1},...,m_{r}=0}^{∞} G(m_{1}, …, m_{r}).

Then

Bos(A) = 1/Ferm(A).

When we specialize to q = 1, then Theorem 1 becomes the (2), which explains why our result is a q-version of the MacMahon Master Theorem. For a motivation of Theorem 1, see Section 3.

The above result is not only interesting from the combinatorial point of view, but it is also a key ingredient in a finite noncommutative formula for the colored Jones function of a knot. This will be explained in a subsequent publication, [GL].

1.4. Computer code. The results of the paper have been verified by computer code, written by the third author. Maple programs QuantumMACMAHON and qMM are

available at: http://www.math.rutgers.edu/

∼zeilberg/. The former proves rigorously Theorem 1 for any fixed r.

1.5. Acknowledgement. The authors wish to thank the anonymous referee who pointed out an error in an earlier version of the paper, and Martin Loebl for enlightening conversations.

2. Proof

2.1. Some lemmas on operators. The proof will make crucial use of a calculus of difference operators, developed by the third author in [Z1]. This calculus of difference operators predates the more advanced calculus of holonomic functions, developed by the third author in [Z2].

Difference operators act on discrete functions F, that is functions whose domain is ℕ^{r}. For example, consider the shift-operators M_{i} and the multiplication operator Q_{i} which act on a discrete function F (m_{1}, …, m_{r}) by

(M_{i}F)(m_{1}, …, m_{r}) := F (m_{1}, …, m_{i−1}, m_{i} + 1, m_{i+1}, …, m_{r})
(Q_{i}F)(m_{1}, …, m_{r}) := q^{m_{i}}F (m_{1}, …, m_{r}).

It is easily seen that

M_{i}Q_{i} = qQ_{i}M_{i}.

Abbreviating Q_{i} by q^{m_{i}}, we obtain that:

(6)  M_{i}q^{m_{i}} = q^{m_{i}+1}M_{i}  M_{i}q^{m_{j}} = q^{m_{j}}M_{i} for i ≠ j.

Another example is the operator x̂_{i} which left multiplies F by x_{i}. Notice that x̂_{j}x̂_{i} = qx̂_{i}x̂_{j} for j > i. In the proof below, we will denote x̂_{i} by x_{i}. In that case, the identity x_{j}x_{i} = qx_{i}x_{j} for j > i holds in the quantum algebra, as well as in the algebra of operators.

Before embarking on the proof, we need the following readily-verified lemmas. Lemma 2.1. (commuting X_{i} with X_{j}) For 1 ≤ i < j ≤ r, X_{j}X_{i} = qX_{i}X_{j}. Lemma 2.2. (commuting x_{i} with X_{j}) For each of the a_{ij}, define the operator Q_{ij} acting on expressions P involving a_{ij} by Q_{ij}P (a_{ij}) := P (qa_{ij}). Then, for any 1 ≤ i, j ≤ r, and integer m_{i} and any expression F

x_{i}^{−m_{i}}X_{j}F = [(Q_{j1}^{−1}Q_{j2}^{−1} ⋯ Q_{j,i−1}^{−1}Q_{j,i+1} ⋯ Q_{jr})^{m_{i}}X_{j}]x_{i}^{−m_{i}}F.

Lemma 2.3. (Column expansion with respect to the last column): Given an r by r matrix (a_{ij}) (not necessarily quantum) let A_{i} be the minor of the entry a_{ir}, i.e. the r − 1 by r − 1 matrix obtained by deleting the i^{th} row and r^{th} column. Then

det_{q}(A) = ∑_{i=1}^{r}(−q)^{i−r} (det_{q}A_{i}) a_{ir}.

Lemma 2.4. If A is a matrix that satisfies Equation (5) and A′ denotes a matrix obtained by interchanging the i and j columns columns of A, then det_{q}(A′) = (−q)^{−inv(ij)}det_{q}(A).

Proof. Suppose first that we interchange two adjacent colums i and j := i + 1. Consider the involution of S_{r} that sends a permutation π to π′ = π(ij). Given π ∈ S_{r}, let (A; π) = (−1)^{inv(π)}a_{π_{1}1} … a_{π_{r}r} denote the contribution of π in det_{q}(A). Then, det_{q}(A) = ∑_{π}(A; π). Equation (5) implies that

(A; π) + (A; π′) = (−q)((A′; π) + (A′; π′)).

Summing over all permutations proves the result when j = i + 1.

Observe that when j = i + 1, the matrix A′ is no longer right-quantum since it does not satisfy (5). However, the proof used only the fact that (5) holds for the i and i + 1 columns of A.

Thus, the proof can be iterated inv(ij) times to commute the i and j > i columns of A. The result follows. □

Lemma 2.5. (Equal columns imply that det_{q} vanishes): Let A be a right-quantum matrix. In the notation of Lemma 2.3, for all j ≠ r,

∑_{i=1}^{r}(−q)^{i−r} (det_{q}A_{i}) a_{ij} = 0.

Proof. If j = r − 1, it is easy to see that q-commutation along the entries in every column of A imply that the sum vanishes.

If j < r − 1, use Lemma 2.4 to reduce it to the case of j = r − 1. □

Remark 2.6. One can give an alternative proof of Lemmas 2.4 and 2.5 from the trivial 2 by 2 case and, by induction using the q-Laplace expansion of a q-determinant that is completely analogous to the classical case.

2.2. Proof of Theorem 1. The proof is a quantum-adaptation of the “operatorelimination” proof of MacMahon’s Master Theorem given in [Z1]. Fix a rightquantum matrix A.

Observe that G(m_{1}, …, m_{r}) is the coefficient of x_{1}^{0} … x_{r}^{0} in

H(m_{1}, …, m_{r}; x_{1}, …, x_{r}) := x_{r}^{−m_{r}} ⋯ x_{2}^{−m_{2}}x_{1}^{−m_{1}} ∏_{i=1}^{r} X_{i}^{m_{i}}.

We will think of H as a discrete function, that is as a function of (m_{1}, …, m_{r}) ∈ ℕ^{r}. H takes values in the ring of noncommutative Laurrent polynomials in the x_{i}s, with coefficients in the ring generated by the entries of A, modulo the ideal given by (3)- (5).

Let’s see how the shift operators M_{i} acts on H. By definition,

M_{i}H(m_{1}, …, m_{r}; x_{1}, …, x_{r}) =

x_{r}^{−m_{r}} ⋯ x_{i+1}^{−m_{i+1}}x_{i}^{−m_{i}−1}x_{i−1}^{−m_{i−1}} ⋯ x_{1}^{−m_{1}}X_{1}^{m_{1}} ⋯ X_{i−1}^{m_{i−1}}X_{i}^{m_{i}+1}X_{i+1}^{m_{i+1}} ⋯ X_{r}^{m_{r}}. By moving x_{i}^{−1} to the front and X_{i} in front of X_{1}^{m_{1}}, and using Lemma 2.1 and x_{j}x_{i} = qx_{i}x_{j}, we have

M_{i}H(m_{1}, …, m_{r}) = q^{m_{r}+m_{r−1}+···+m_{i+1}−m_{1}−m_{2}−···−m_{i−1}}x_{i}^{−1}[x_{r}^{−m_{r}} ⋯ x_{1}^{−m_{1}}X_{i}]X_{1}^{m_{1}} ⋯ X_{r}^{m_{r}}.

By moving X_{i} next to x_{i}^{−1} and using Lemma 2.2 this equals to:

q^{m_{r}+m_{r−1}+···+m_{i+1}−m_{1}−m_{2}−···−m_{i−1}}x_{i}^{−1}·

[(Q_{i2} ⋯ Q_{ir})^{m_{1}}(Q_{i1}^{−1}Q_{i3} ⋯ Q_{ir})^{m_{2}}(Q_{i1}^{−1}Q_{i2}^{−1}Q_{i4} ⋯ Q_{ir})^{m_{3}} ⋯ (Q_{i1}^{−1}Q_{i2}^{−1} ⋯ Q_{i,r−1}^{−1})^{m_{r}}X_{i}]·

x_{r}^{−m_{r}} ⋯ x_{1}^{−m_{1}}X_{1}^{m_{1}} … X_{r}^{m_{r}},

which is equal to

q^{m_{r}+m_{r−1}+···+m_{i+1}−m_{1}−m_{2}−···−m_{i−1}}x_{i}^{−1}·

(q^{−m_{2}−m_{3}−···−m_{r}}a_{i1}x_{1} + q^{m_{1}−m_{3}−···−m_{r}}a_{i2}x_{2} + ⋯ + q^{m_{1}+m_{2}+···+m_{r−1}}a_{ir}x_{r}) H(m_{1}, …, m_{r}; x_{1}, …, x_{r})

Multiplying out and rearranging, we get that the discrete function H(m_{1}, …, m_{r}; x_{1}, …, x_{r}) is annihilated by the r operators (i = 1, 2, …, r)

P_{i} := ∑_{j=1}^{i−1} −q^{−m_{j}−2m_{j+1}−···−2m_{i−1}−m_{i}}a_{ij}x_{j}+(M_{i}−a_{ii})x_{i}+ ∑_{j=i+1}^{r} −q^{m_{i}+2m_{i+1}+···+2m_{j−1}+m_{j}}a_{ij}x_{j}.

Now comes a nice surprise. Let us define b_{ij} to be the coefficient of x_{j} in P_{i}. For example, for r = 3 we have:

B = (\begin{matrix}M_{1} − a_{11} & −q^{m_{1}+m_{2}}a_{12} & −q^{m_{1}+2m_{2}+m_{3}}a_{13} \\ −q^{−m_{1}−m_{2}}a_{21} & M_{2} − a_{22} & −q^{m_{2}+m_{3}}a_{23} \\ −q^{−m_{1}−2m_{2}−m_{3}}a_{13} & −q^{−m_{2}−m_{3}}a_{23} & M_{3} − a_{33} \end{matrix}) .

Lemma 2.7. B is a right-quantum matrix.

Proof. It is easy to see that the entries in each column of B q-commute. To prove Equation (5), consider the following cases for a 2 by 2 submatrix C of B: C contains two, (resp. one, resp. no) diagonal entries of B, and prove it case by case, using the fact that the operators M_{i} and q^{m_{j}} commute with the a_{ij}, and satisfy the commutation relations (6). □

Now we eliminate x_{1}, x_{2}, …, x_{r−1} by left-multiplying P_{i} by the minor of b_{ir} in B = (b_{ij}) times (−q)^{i−r}, for each i = 1, 2, …, r, and adding them all up. Since B is right-quantum (by Lemma 2.7), Lemma 2.5 implies that the coefficients of x_{1}, …, x_{r−1} all vanish, and det_{q}(B)x_{r}H = 0. After left multiplying by x_{r}^{−1} which commutes with the entries in B, we obtain that

det_{q}(B)H(m_{1}, …, m_{r}; x_{1}, …, x_{r}) = 0.

Since the entries of B do not contain x_{i}’s, it follows that det_{q}(B) annihilates each and every single coefficient of H, in particular its constant term. Taking the constant term yields

det_{q}(B)G(m_{1}, …, m_{r}) = 0.

Here comes the next surprise. Lemma 2.8. (a) We have:

det_{q}(B) = ∑_{J⊂{1,...,r}}(−1)^{|J|}det_{q}(A_{J})MJ̄

where J̄ = {1, …, r} − J and M_{J} = ∏_{j∈J} M_{j}.

(b) In particular,

det_{q}(B)|_{M_{1}=···=M_{r}=1} = Ferm(A).

Proof. Let us expand det_{q}(B) as a sum over permutations π ∈ S_{r}. We have:

det_{q}(B) = ∑_{π∈S_{r}}(−q)^{−inv(π)}b_{π_{1}1}b_{π_{2}2} ⋯ b_{π_{r}r}

= ∑_{π∈S_{r}} ∏_{i=1}^{r}(−q)^{−inv(π,i)}b_{π_{i}i}

where inv(π, i) is the number of j > i such that π_{i} > π_{j}. Now, b_{ij} = q_{ij}a_{ij} for i ≠ j, where q_{ij} is a monomial in the variables q^{m_{i}}, and ∏_{j} q_{ij} = 1. Moreover, the exponent of q^{m_{i}} in q_{ij} is 2 if j > i and −2 if j < i.

Since ∏_{j} q_{ij} = 1, we can move the monomials q_{ij} in the left of ∏_{i}(−q)^{inv(π,i)}b_{π_{1}1} … b_{π_{r}r}, and then cancel them. The monomials commute with all entries of the matrix b_{ij}, except with the diagonal ones. Commuting q^{2m_{i}} with b_{ii} = δ_{π_{i}i}M_{ii} − a_{ii} gives: b_{ii}q^{2m_{i}} = q^{2m_{i}}(δ_{π_{i}i}q^{2}M_{ii} − a_{ii}). In other words, it replaces M_{i} by q^{2}M_{i}. Thus, we have:

det_{q}(B) = ∑_{π∈S_{r}} ∏_{i=1}^{r}(−q)^{−inv(π,i)}(δ_{π_{i}i}q^{2inv(π,i)}M_{i} − a_{π_{i}i})
= ∑_{π∈S_{r}} ∑_{J⊂{1,...,r}} ∏_{i∈J}(−q)^{−inv(π,i)}δ_{π_{i}i}q^{2inv(π,i)}M_{i} ∏_{i∉J}(−q)^{−inv(π,i)}(−a_{π_{i}i})

Now, rearrange the summation. Observe that every permutation π of {1, …, r} gives rise to a permutation π′ on the set {1, …, r} − Fix(π), where Fix(π) is the fixed point set of π. Moreover, inv(π′, i) = inv(π, i) − |{j ∈ J : j > i}|. Using this, part (a) follows. Part (b) follows from part (a) and the definition of Ferm(A). □

Hence

∑_{J⊂{1,...,r}}(−1)^{|J|}det_{q}(A_{J})MJ̄G(m_{1}, …, m_{r}) = 0.

Summing over ℕ^{r}, we get:

∑_{m_{1},...,m_{r}=0}^{∞} ∑_{J⊂{1,...,r}}(−1)^{|J|}det_{q}(A_{J})MJ̄G(m_{1}, …, m_{r}) = 0.

For a subset J = {k_{1}, …, k_{j}} of {1, …, r}, we denote by G_{J}(m_{k_{1}}, …, m_{k_{j}}) the evalutation G(m_{1}, …, m_{r}) at m_{i} = 0 for all i ∉ J, and we define

S_{J} = ∑_{m_{k_{1}} ,...,m_{k_{j}} =0}^{∞} G(m_{1}, …, m_{r}).

Using telescoping cancellation, the inclusion-exclusion principle, and Lemma 2.8(b), the above equation becomes

∑_{J⊆{1,...,r}}(−1)^{|J|}Ferm(A_{J})S_{J} = 0.

Using induction (with respect to r), together with S_{∅} = 1, we obtain that Ferm(A)S_{{1,...,r}} = 1. This concludes the proof of the theorem. □

3. Some remarks on the boson-fermion correspondence

Let us give some motivation for Theorem 1 from the point of view of quantum topology.

For a reference on quantum space and quantum algebra, see [Ka, Chapter IV] and [M].

Recall that a vector (column or row) of r indeterminate entries x_{1}, …, x_{r} lies in r-dimensional quantum space A^{r|0} iff its entries satisfy

x_{j}x_{i} = qx_{i}x_{j}

for all 1 ≤ i < j ≤ r.

Recall that a right (resp. left) endomorphism of A^{r|0} is a matrix A = (a_{ij}) of size r whose entries commute with the coordinates x_{i} of a vector x = (x_{1}, …, x_{r})^{T} ∈ A^{r|0} and in addition, Ax (resp. x^{T} A) lie in A^{r|0}. Recall also that an endomorphism of A^{r|0} is one that is right and left endomorphism.

It is easy to see (eg. in [Ka, Thm. IV.3.1]) that A is a right-quantum (i.e., a

right-endomorphism) iff for every 2 by 2 submatrix (\begin{matrix}a & b \\ c & d \end{matrix}) of A we have:

ca = qac,  db = qbd,  ad = da + q^{−1}cb − qbc.

Moreover, A is left-quantum iff for every 2 by 2 submatrix of A (as above) we have:

ba = qab,  dc = qcd,  ad = da + q^{−1}bc − qcb.

Finally, A is quantum iff for every 2 by 2 submatrix of A (as above) we have: (7) ba = qab, ca = qac, db = qbd, dc = qcd, cb = bc, ad = da + q^{−1}cb − qbc.

The set of quantum matrices A are the points of the r-dimensional quantum algebra M_{q}(r), which is defined to be the quotient of the free algebra in noncommuting variables x_{ij} for 1 ≤ i, j, ≤ r, modulo the left ideal generated by the commutation relations of Equation (7).

The algebra M_{q}(r) has interesting and important structure. M_{q}(r) is Noetherian, has no zero divisors, and in addition, a basis for the underlying vector space is given by the set of sorted monomials {∏_{i,j} a_{ij}^{n_{ij}}|n_{ij} ≥ 0} where the product is taken lexicographically; see [Ka, Thm IV.4.1]. An important quotient of M_{q}(r) is the quantum group SL_{q}(r) := M_{q}(r)/(det_{q} − 1), which is a Hopf algebra [Ka, Sec.IV.6] whose representation theory gives rise to the quantum group invariants of knots, such as the celebrated Jones polynomial.

Observing that

tr S^{n}(A) = ∑_{m_{1}+...m_{r}=n} G(m_{1}, …, m_{r})
tr Λ^{n}(A) = = ∑_{J⊂{1,...,r},|J|=n} det_{q}(A_{J})

Theorem 1 implies that: Theorem 2. If A is in M_{q}(r), then

\frac{1}{Ferm(A)} = ∑_{n=0}^{∞} tr S^{n}(A)

Since the algebra M_{q}(r) has a vector space basis given by sorted monomials, it should be possible to give an alternative proof of the quantum MacMahon Master Theorem using combinatorics on words, as was done in [FZ] for several proofs of the MacMahon Master theorem. We hope to return to this alternative point of view in the near future.

References

[A] G. E. Andrews, Problems and prospects for basic hypergeometric functions, The Theory and Applications of Special Functions (R. Aseky, Editor), Academic Press, New York, 1975, 191-224.

[FRT] L. Fadeev, N. Reshetikhin and L. Takhtadjian, Quantization of Lie groups and Lie algebras, Leningrad Math. Journal 1 (1990), 193-225.

[FZ] D. Foata and D. Zeilberger, A combinatorial proof of Bass’s evaluation of the Ihara-Selberg zeta function for graphs, Transactions Amer. Math. Soc. 351 (1999) 2257–2274.

[GL] S. Garoufalidis and TTQ Le, in preparation.

[Ka] C. Kassel, Quantum groups, GTM 155 Springer-Verlag (1995).

[MM] P. A. MacMahon, Combinatory Analysis vol. 1, Cambridge University Press, 1917. Reprinted by Chelsea, 1984.

[M] Y. Manin, Quantum groups and noncommutative geometry, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1988.

[Z1] D. Zeilberger, The algebra of linear partial difference operators and its applications, SIAM J. Math. Anal. 11 (1980) 919–934.

[Z2] D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (1990) 321–368.

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160,

USA, http://www.math.gatech
.edu/∼stavros
   E-mail address: stavros@math.gatech.edu

Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

   E-mail address: letu@math.gatech.edu

Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019, USA

   E-mail address: zeilberg@math.rutgers.edu