Non-commutative Extensions of the Macmahon Master Theorem

We present several non-commutative extensions of the MacMahon Master Theorem, further extending the results of Cartier-Foata and Garoufalidis-LêZeilberger. The proofs are combinatorial and new even in the classical cases. We also give applications to the β-extension and Krattenthaler-Schlosser’s q-analogue. Introduction The MacMahon Master Theorem is one of the jewels in enumerative combinatorics, and it is as famous and useful as it is mysterious. Most recently, a new type of algebraic generalization was proposed in [GLZ] and was further studied in [FH1, FH2, FH3, HL]. In this paper we present further generalizations of the MacMahon Master Theorem and several other related results. While our generalizations are algebraic in statement, the heart of our proofs is completely bijective, unifying all generalizations. In fact, we give a new bijective proof of the (usual) MacMahon Master Theorem, modulo some elementary linear algebra. Our approach seems to be robust enough to allow further generalizations in this direction. Let us begin with a brief outline of the history of the subject. The Master Theorem was discovered in 1915 by Percy MacMahon in his landmark two-volume “Combinatory Analysis”, where he called it “a Master Theorem in the Theory of Partitions” [MM, page 98]. Much later, in the early sixties, the real power of Master Theorem was discovered, especially as a simple tool for proving binomial identities (see [GJ]). The proof of the MacMahon Master Theorem using Lagrange inversion is now standard, and the result is often viewed in the analytic context [Go, GJ]. An algebraic approach to MacMahon Master Theorem goes back to Foata’s thesis [F1], parts of which were later expanded in [CF] (see also [L]). The idea was to view the theorem as a result on “words” over a (partially commutative) alphabet, so one can prove it and generalize it by means of simple combinatorial and algebraic considerations. This approach became highly influential and led to a number of new related results (see e.g. [K, Mi, V, Z]). While the Master Theorem continued to be extended in several directions (see [FZ, KS]), the “right” qand non-commutative analogues of the results evaded discovery until recently. This was in sharp contrast with the Lagrange inversion, whose qand non-commutative analogues were understood fairly well [Ga, GaR, Ge, GS, Kr, PPR, Si]. Unfortunately, no reasonable generalizations of the Master Theorem followed from these results. Date: July 28, 2006. 1 2 MATJAŽ KONVALINKA AND IGOR PAK An important breakthrough was made by Garoufalidis, Lê and Zeilberger (GLZ), who introduced a new type of q-analogue, with a puzzling algebraic statement and a technical proof [GLZ]. In a series of papers, Foata and Han first modified and extended the Cartier-Foata combinatorial approach to work in this algebraic setting, obtaining a new (involutive) proof of the GLZ-theorem [FH1]. Then they developed a beautiful “1 = q” principle which gives perhaps the most elegant explanation of the results [FH2]. They also analyze a number of specializations in [FH3]. Most recently, Hai and Lorenz gave an interesting algebraic proof of the GLZ-theorem, opening yet another direction for exploration (see Section 13). This paper presents a number of generalizations of the MacMahon Master Theorem in the style of Cartier-Foata and Garoufalidis-Lê-Zeilberger. Our approach is bijective and is new even in the classical cases, where it is easier to understand. This is reflected in the structure of the paper: we present generalizations one by one, gradually moving from well known results to new ones. The paper is largely self-contained and no background is assumed. We begin with basic definitions, notations and statements of the main results in Section 1. The proof of the (usual) MacMahon Master Theorem is given in Section 2. While the proof here is elementary, it is the basis for our approach. A straightforward extension to the Cartier-Foata case is given in Section 3. The right-quantum case is presented in Section 4. This is a special case of the GLZ-theorem, when q = 1. Then we give a q-analogue of the Cartier-Foata case (Section 5), and the GLZ-theorem (Section 6). The subsequent results are our own and can be summarized as follows: • The Cartier-Foata (qij)-analogue (Section 7). • The right-quantum (qij)-analogue (Section 8). • The super-analogue (Section 9). • The β-extension (Section 10). The (qij)-analogues are our main result; one of them specializes to the GLZ-theorem when all qij = q. The super-analogue is a direct extension of the classical MacMahon Master Theorem to commuting and anti-commuting variables. Having been overlooked in previous investigations, it is a special case of the (qij)-analogue, with some qij = 1 and others = −1. Our final extension is somewhat tangential to the main direction, but is similar in philosophy. We show that our proof of the MacMahon Master Theorem can be easily modified to give a non-commutative generalization of the so called β-extension, due to Foata and Zeilberger [FZ]. In Section 11 we present one additional observation on the subject. In [KS], Krattenthaler and Schlosser obtained an intriguing q-analogue of the MacMahon Master Theorem, a result which on the surface does not seem to fit the above scheme. We prove that in fact it follows from the classical Cartier-Foata generalization. As the reader shall see, an important technical part of our proof is converting the results we obtain into traditional form. This is basic linear algebra in the classical case, but in non-commutative cases the corresponding determinant identities are either less known or new. For the sake of completeness, we present concise proofs of all of them in Section 12. We conclude the paper with final remarks and open problems. NON-COMMUTATIVE MACMAHON MASTER THEOREM 3 1. Basic definitions, notations and main results 1.1. Classical Master Theorem. We begin by stating the Master Theorem in the classical form: Theorem 1.1. (MacMahon Master Theorem) Let A = (aij)m×m be a complex matrix, and let x1, . . . , xm be a set of variables. Denote by G(k1, . . . , km) the coefficient of x1 1 · · · x km m in