A basis for the right quantum algebra and the 1 = q principle

Transcription

1 J Algebr Comb 008 7: 6 7 DOI 0.007/s A basis for the right quantum algebra and the q principle Dominique Foata Guo-Niu Han Received: 6 January 007 / Accepted: May 007 / Published online: 6 June 007 Springer ScienceBusiness Media, LLC 007 Abstract We construct a basis for the right quantum algebra introduced by Garoufalidis, Lê and Zeilberger and give a method making it possible to go from an algebra subject to commutation relations without the variable q to the right quantum algebra by means of an appropriate weight-function. As a consequence, a strong quantum MacMahon Master Theorem is derived. Besides, the algebra of biwords is systematically in use. Keywords Right quantum algebra Quantum MacMahon master theorem Mathematics Subject Classification 000 Primary 05A9 05A0 6W5 7B7 Introduction In their search for a natural q-analogue of the MacMahon Master Theorem, Garoufalidis et al. [4] have introduced the right quantum algebra R q defined to be the associative algebra over a commutative ring K, generated by r elements X xa x,a r, r, subject to the following commutation relations: X yb X xa X xa X yb q X xb X ya qx ya X xb, X ya X xa q X xa X ya, x >y, all a x>y,a>b D. Foata Institut Lothaire, rue Murner, Strasbourg, France foata@math.u-strasbg.fr G.-N. Han I.R.M.A. UMR 750, Université Louis Pasteur, 7 rue René-Descartes, Strasbourg, France guoniu@math.u-strasbg.fr

2 64 J Algebr Comb 008 7: 6 7 with q being an invertible element in K. The right quantum algebra in the case r has already been studied by Rodríguez-Romo and Taft [], who set up an explicit basis for it. On the other hand, a basis for the full quantum algebra has been duly constructed see [, Theorem.5., p. 8] for an arbitrary r. It then seems natural to do the same with the right quantum algebra for each r. This is the first goal of the paper. In fact, the paper originated from a discussion with Doron Zeilberger, when he explained to the first author how he verified the quantum MacMahon Master identity for each fixed r by computer code. His computer program uses the fact, as he is perfectly aware, that the set of irreducible biwords which will be introduced in the sequel generates the right quantum algebra. For a better understanding of his joint paper [4] and also for deriving the q principle, it seems essential to see whether the set of irreducible biwords has the further property of being a basis, and it does. Thanks to this result, a strong quantum MacMahon Master Theorem can be subsequently derived. When manipulating the above relations, the visible part of the commutations is made within the subcripts of the X xa s, written as such. It is then important to magnify them by having an adequate notation. To this end, we replace each X xa by the biletter x a, so that, in further computations, products X x,a X x,a X xl,a l become biwords x x x l a a a l, objects that have been efficiently used in combinatorial contexts [9, 0]. Accordingly, the commutation rules for the right quantum algebra can be written as yx ba xy ab q xy ba q yx ab, x>y, a>b yx aa q xy aa, x >y, all a. We shall use the following notation. The positive integer r will be kept fixed throughout and A will denote the alphabet {,,...,r}. Abiword on A is a n matrix α x x n a a n, n 0, whose entries are in A, the first resp. second row being called the top word resp. bottom word ofthebiwordα. The number n is the length of α we write lα n. The biword α can also be viewed as a word of biletters x a xn a n, those biletters xi a i being pairs of integers written vertically with x i,a i A for all i,...,n.theproduct of two biwords is their concatenation. Let B denote the set of biletters and B the set of all biwords. Let α B such that lα and i lα. The biword α can be factorized as α β xy ab γ, where x,y,a,b A and β,γ B with lβ i. Say that α has a double descent at position i if x>yand a b. Notice the discrepancy in the inequalities on the top word and the bottom one. A biword α without any double descent is said to be irreducible. The set of all irreducible biwords is denoted by B irr. Let Z be the ring of all integers, and let Z[q,q ] be the ring of the polynomials in the variables q, q subject to the rule qq with integral coefficients. The set A Z B of the formal sums α cαα, where α B and cα Z, together with the above biword multiplication, the free addition and the free scalar product, forms an algebra over Z, called the free biword large Z-algebra. The formal sums

3 J Algebr Comb 008 7: α cαα will be called expressions. An expression α cαα is said to be irreducible if cα 0 for all α B irr. The set of all irreducible expressions is denoted by A irr. Similarly, let A q Z[q,q ] B denote the large Z[q,q ]-algebra of the formal sums α cαα, where cα Z[q,q ] for all α B. Following Bergman s method [] we introduce two reduction systems S and S q as being the sets of pairs α, [α] B A and α, [α] q B A q, respectively, defined by xy [ ab, xy ] [ ab with xy ] ab yx ba yx ab xy ba, x >y, a >b S xy [ aa, xy ] [ aa with xy ] aa yx aa, x >y, all a. xy [ ab, xy ] [ ab q with xy ab ]q yx ba q yx ab q xy ba, x >y, a >b S q xy [ aa, xy ] [ aa q with xy aa ]q q yx aa, x >y, all a. Notice that the equations xy [ ab xy ab ]q 0, xy [ aa xy ] aa 0 are simple rewritings q of the commutation rules of the right quantum algebra and that S is deduced from S q by letting q. Let I resp. I q be the two-sided ideal of A resp. of A q generated by the elements γ [γ ] resp. γ [γ ] q such that γ, [γ ] S resp. γ [γ ] q S q. The quotient algebras R A/I and R q A q /I q are called the -right quantum algebra and the q-right quantum algebra, respectively. In Sect. we define a Z-linear mapping E [E] of A onto the Z-module A irr of the irreducible expressions, called reduction. Using Bergman s Diamond Lemma [] this reduction will serve to obtain a model for the algebra R, as stated in the next theorem. Theorem A set of representatives in A for R isgivenbythez-module A irr. The algebra R may be identified with the Z-module A irr, the multiplication being given by E F [E F ] for any two irreducible expressions E, F. With this identification B irr is a basis for R. Four statistics counting various kinds of inversions will be needed. If α u v x x x n a a a n is a biword, let inv u # { } i, j i<j n, x i >x j imv v # { } i, j i<j n, a i a j inv α inv v inv u inv α imv v inv u. The first resp. second statistic inv resp. imv is the usual number of inversions resp. of large inversions of a word. Notice that inv may be negative. The weight function φ defined for each biword α by φα q inv α α can be extended to all of A q by linearity. Clearly, it is a Z[q,q ]-module isomorphism of A q onto itself. The second result of the paper is stated next.

4 66 J Algebr Comb 008 7: 6 7 Theorem The Z[q,q ]-module isomorphism φ of A q onto itself induces a Z[q,q ]-module isomorphism φ of A q /I onto A q /I q. In particular, R q has the same basis as R. Now, a circuit is defined to be a biword whose top word is a rearrangement of the letters of its bottom word. The set of all circuits is denoted by C. It is clearly a submonoid of B. An expression E α cαα is said to be circular if cα 0for all α C. Clearly, the sum and the product of two circular expressions is a circular expression, so that the set of circular expressions in A resp. in A q is a subalgebra of A resp. of A q, which will be denoted by resp. by q. Theorem The restriction of the Z[q,q ]-module isomorphism φ of Theorem to q /I is a Z[q,q ]-algebra isomorphism onto q /I q. can q q /I φ q module-iso can φ q /I q algebra-iso A q φ module-iso A q can can A q /I φ algebra-iso A q /I q Accordingly, each identity holding in q /I has an equivalent counterpart in q /I q. This is the q principle. An illustration of this principle is given by the qmm Theorem derived by Garoufalidis et al. [4]. Those authors have introduced the q-fermion as being the sum Fermq J q inv σ σi σ i σi l, i i i l J A σ S J where S J is the permutation group acting on the set J {i <i < <i l }, and the q-boson as the sum Bosq w q inv w w w over all words w from the free monoid A generated by A, where w stands for the nondecreasing rearrangement of the word w. Theq-Fermion and q-boson belong to A q. Garoufalidis et al. have proved [4], see also [] for another proof the following identity Fermq Bosq mod I q. This identity may aprioribe regarded as a q-version of the MacMahon Master Theorem, as it reduces to the classical MacMahon s identity [, pp. 9 98], when q and the biletters are supposed to commute. By the q principle Theorem, we obtain the following result, which shows that the variable q is in fact superfluous.

5 J Algebr Comb 008 7: Corollary 4 The identity Fermq Bosq mod I q holds if and only if Ferm Bos mod I holds. When applying Theorem to the quantum MacMahon Master Theorem, we obtain the following result, which can be regarded as a strong quantum MacMahon Master Theorem. Corollary 5 The following identity holds: [ Ferm Bos ]. The proof of Theorem, given in Sect., is based on Bergman s Diamond Lemma [, Theorem.]. It consists of verifying that the conditions required by the Diamond Lemma hold in the present situation. Theorems and areprovedin Sect. and Corollaries 4 and 5 in the last section. The reduction process For applying Bergman s Diamond Lemma [] we need a so-called reduction system that satisfies two conditions: the descending chain condition holds its ambiguities are resolvable. Those terms will be explained shortly. Let Z B be the subalgebra of A of all sums α cαα α B, where only a finite number of the cα s are non-zero. The elements of Z B will be called finite expressions. The reduction system for Z B is simply the set S of pairs α, [α] B Z B, already described in Sect.. By means of this reduction system we introduce the reduction itself as being the mapping E [E] of Z B onto the set Z B irr of the finite irreducible expressions defined by the following axioms: C The reduction is linear, i.e. for E,E Z B, [c E c E ]c [E ]c [E ]. C [E]E for every E Z B irr. C For xy ab Birr,i.e.x>yand a b, then [ xy ] C. ab yx ba yx ab xy ba if a>b ] C. yx if a b. [ xy aa C4 Let α B irr be a reducible biword. For each factorization α β xy ab γ, where x,y,a,b A, β x a Birr and γ B,wehave aa [α] [ β [ xy ab] γ ]. Although the reduction is recursively defined, it is well defined. Every time Condition C is applied, the running biword α is transformed into either three new biwords C. or a new biword C.. The important property is the fact that the

6 68 J Algebr Comb 008 7: 6 7 statistic inv of each of the new biwords is strictly less than inv α. Thus, after finitely many successive applications of C, an irreducible expression is derived. When the biword α has more than one double descent, Condition C4 says that Condition C must be applied at the first double descent position. Consequently, the final irreducible expression is unique. There are several other ways to map each expression onto an irreducible expression, using relations C. and C.. The reduction defined above is only one of those mappings. The important feature is the fact that such a mapping involves finitely many applications of relations C. and C.. Following Bergman we say that Condition the descending chain condition holds for the pair Z B,S. Now an expression E is said to be reduction-unique under S if the irreducible expression derived from E does not depend on where the applications of C. and C. take place. More formally, each expression is reduction-unique under S if for every biword α and every factorization α βα γβ,α,γ B we have Reduction-unique [α][β[α ]γ ]. Now examine the second condition required by the Diamond Lemma. There is an ambiguity in S if there are two pairs α, [α] and α, [α ] in S such that α βγ, α γδ for some nonempty biwords β,γ,δ B. This ambiguity is said to be resolvable if [[α]δ] [β[α ]], i.e.[[βγ]δ] [β[γδ]]. In our case, this means that β x a, γ y b, δ z c with x>y>zand a b c. Using the first three integers instead of the letters x,y,z,a,b,c there are four cases to be studied: i β, γ, δ ii β, γ, δ iii β, γ, δ iv β, γ, δ. Accordingly, the ambiguities in S are resolvable if the following four identities hold: [[ [[ [[ ] ] [ ] ] [ [ [ [ ]]. ]]. [ ] ] ]]. [[ [ ] ] [ ]]..4 Let us prove., the other three identities being handled in a similar manner. Proof of. For an easy reading of the coming calculations we have added subscripts A,B,...,J to certain brackets, which should help spot the subscripted brackets in the various equations. The left-hand side is evaluated as follows: [[ ] [[ ] ] ]A [[ ] ]B [[ ] ] C

7 J Algebr Comb 008 7: [[ ] ]A [ [ [ ]] [ ]]D [ [ ]]I [ [ ]] J [[ ] ]B [ [ [ ]] [ ]]E [ [ ]]F [ [ ]] [[ ] ]C [ [ [ ]] [ ]]G [ [ [ ]] [ ]] H [ [ ]]D [[ ] [[ ] ] [[ ] ] [[ ] ] ] [ [ ]]E [[ ] [[ ] ] [[ ] ] [[ ] ] ] [ [ ]]F [[ ] [[ ] ] [[ ] ] [[ ] ] ] I [ [ ]]G [[ ] [[ ] ] [[ ] ] [[ ] ] ] [ [ ]]H [[ ] [[ ] ] [[ ] ] ]J [[ ] ]. The sum of the above nine equalities yields [[ ] ]. As for the right-hand side we have [ [ [ ]] [ ]]A [ [ ]]B [ [ ]] C [[ ] ]A [[ ] ]D [[ ] ]I [[ ] ] J [[ ] ]B [[ ] ]E [[ ] ]F [[ ] ] [[ ] ]C [[ ] ]G [[ ] [[ ] ] ] [ [ ]]D [ [ [ ]] [ [ ]] [ ]] [ [ ]]E [ [ [ ]] [ [ ]] [ ]] [ [ ]]F [ [ [ ]] [ [ ]] [ ]] I [ [ ]]G [ [ [ ]] [ [ ]] [ ]] [ [ ]]H [ [ [ ]] [ ]]J [ [ ]]. The sum of the above nine equalities yields [ ]] [ [[ ] ] left-hand side. H

8 70 J Algebr Comb 008 7: 6 7 Let I be the two-sided ideal of Z B generated by the elements γ [γ ] such that γ, [γ ] S, and let R be the quotient Z B /I. As the pair Z B,S satisfies the descending chain condition and since the ambiguities are resolvable, Bergman s Diamond Lemma implies the following theorem. Theorem 6 A set of representatives in Z B for R is given by the Z-module Z B irr. The algebra R may be identified with Z B irr, the multiplication being given by E F [E F ] for any two finite irreducible expressions E, F. With this identification B irr is a basis for R. For the proof of Theorem we use the following argument. For each n 0let A n be the n-th degree homogeneous subspace of A consisting of all expressions α cαα such that lα 0iflα n [6, I.6]. As the starting alphabet A is finite, all expressions in A n are finite sums, so that A n Z B for all n. Now, let each expression E from A be written as the sum E n 0 E n with E n A n. As both ideals I and I are generated by expressions from the second degree homogeneous space A,wehaveE 0 mod I if and only if E n 0 mod I for every n 0. In particular, [E n ] also belongs to A n. Theorem follows from Theorem 6 by taking the definition [E] n 0[ E n ]. The q principle Let E be an expression in A q. Theorem is equivalent to saying that the identity E I holds if and only if φe I q. First, we prove the only if part. Because φ is linear, it suffices to consider all linear generators of I, which have the following form: E α rs ii β α sr ii β and E α rs ij β α sr ji β α sr ij β α rs ji β, where α, β are biwords and r, s, i, j are integers such that r<s, i<j.letk inv α rs ij β, then inv α sr ji so that β k, inv α sr φe q k α rs ij ij β k, inv α rs ji β k, β α sr β q α sr ji ij β qα rs ji β Iq. InthesamewayφE I q. The if part can be proved in the same manner.

9 J Algebr Comb 008 7: For Theorem it suffices to prove the identity φef φeφf for any two circular expressions E and F.Asφ is linear, it suffices to do it when E u v, F u v are two circuits. Let invu, u denote the number of pairs x, y such that x resp. y is a letter of u resp. of u and x>y. Since uu vv is a circuit, we have so that inv uu inv u inv u invu, u inv vv inv v inv v invv, v invu, u invv, v inv EF inv vv inv uu inv v inv v inv u inv u inv E inv F. 4 The quantum MacMahon Master Theorem For proving Corollary 4 we apply Theorem to Ferm Bos. Since Ferm and Bos are both circular expressions, the relation Ferm Bos mod I is equivalent to φ Ferm φ Bos mod I q. Finally, it is straightforward to verify Fermq φ Ferm and Bosq φ Bos. Now, to prove Corollary 5 we start with Garoufalidis et al. s result for q, which states that Ferm Bos mod I. But by Theorem we have E Fmod I if and only if [E][F ]. Hence [Ferm Bos][]. 5 Concluding remarks It is worth noticing that Rodríguez and Taft have introduced two explicit left quantum groups for r in[] and [4]. As mentioned in the introduction, the former one has been extended to an arbitrary dimension r by Garoufalidis et al. [4] and has been given an explicit basis in the present paper, while the latter one has been recently modeled after SL q r by Lauve and Taft [8], also for each r. Since the paper has been submitted for publication, several works about the quantum MacMahon Master Theorem have been completed. Hai and Lorenz [5] have derived the identity using Koszul complex techniques. Konvalinka and Pak [7] have

10 7 J Algebr Comb 008 7: 6 7 worked out a several basis q ij -analogue of the quantum MacMahon Master Theorem and extended the q principle developed in our paper to the case q ij. Etingof and Pak [] have derived an extension taking up again the Koszul duality to obtain a new identity having several enumerative applications. Acknowledgements We should like to thank Doron Zeilberger and Stavros Garoufalidis for several fruitful discussions. We are grateful to Christian Kassel, who gave us the references to the book by Parshall and Wang [], the two papers by Rodríguez and Taft [, 4] and the fundamental paper by Bergman []. Finally, we have greatly benefited from Jean-Pierre Jouanolou s ever-lasting mathematical expertise. The authors should like to thank the two referees for careful reading and knowledgeable remarks. References. Bergman, G. M The diamond lemma for ring theory. Advances in Mathematics, 9, Etingof, P., & Pak, I. An algebraic extension of the MacMahon Master Theorem. Proceedings of the American Mathematical Society, to appear.. Foata, D., & Han, G.-N A new proof of the Garoufalidis Lê-Zeilberger quantum MacMahon Master Theorem. Journal of Algebra, 07, Garoufalidis, S., Lê, T.T.Q., & Zeilberger, D The quantum MacMahon Master Theorem. Proceedings of the National Academy of Science of the United States of America, 0, Hai, P. H., & Lorenz, M. Koszul algebras and the quantum MacMahon Master Theorem. The Bulletin of the London Mathematical Society, to appear. 6. Kassel, C Quantum groups. Graduate texts in mathematics Vol. 55. New York: Springer. 7. Konvalinka, M., & Pak, I. Non-commutative extensions of the MacMahon Master Theorem. Advances in Mathematics, to appear. 8. Lauve, A., & Taft, E. J A class of left quantum groups modeled after SL q r. Journal of Pure and Applied Algebra, 08, Lothaire, M. 98. Combinatorics on words. Encyclopedia of mathematics and its applications Vol. 7. London: Addison-Wesley. 0. Lothaire, M. 00. Algebraic combinatorics on words. Encyclopedia of mathematics and its applications Vol. 90. Cambridge: Cambridge Univ. Press.. MacMahon, P. A. 95. Combinatory analysis, Vol.. Cambridge Univ. Press. Reprinted by Chelsea Publ. Co., New York, Parshall, B., & Wang, J.-P. 99. Quantum linear groups. Memoirs of the American Mathematical Society Vol Rodríguez-Romo, S., & Taft, E. 00. Some quantum-like Hopf algebras which remain noncommutative when q. Letters in Mathematical Physics, 6, Rodríguez-Romo, S., & Taft, E A left quantum group. Journal of Algebra, 86,