Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 002, 172 pages 1 On Some Quadratic Algebras I 2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss{Catalan, Universal Tutte and Reduced Polynomials Anatol N. KIRILLOV yzx y Research Institute of Mathematical Sciences (RIMS), Kyoto, Sakyo-ku 606-8502, Japan E-mail: [email protected] URL: http://www.kurims.kyoto-u.ac.jp/~kirillov/ z The Kavli Institute for the Physics and Mathematics of the Universe (IPMU), 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan x Department of Mathematics, National Research University Higher School of Economics, 7 Vavilova Str., 117312, Moscow, Russia Received March 23, 2015, in final form December 27, 2015; Published online January 05, 2016 http://dx.doi.org/10.3842/SIGMA.2016.002 Abstract. We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang{Baxter equations. Key words: braid and Yang{Baxter groups; classical and dynamical Yang{Baxter relations; classical Yang{Baxter, Kohno{Drinfeld and 3-term relations algebras; Dunkl, Gaudin and Jucys{Murphy elements; small quantum cohomology and K-theory of flag varieties; Pieri rules; Schubert, Grothendieck, Schr¨oder,Ehrhart, Chromatic, Tutte and Betti polynomials; reduced polynomials; Chan{Robbins{Yuen polytope; k-dissections of a convex (n + k + 1)- gon, Lagrange inversion formula and Richardson permutations; multiparameter deforma- tions of Fuss{Catalan and Schr¨oderpolynomials; Motzkin, Riordan, Fine, poly-Bernoulli and Stirling numbers; Euler numbers and Brauer algebras; VSASM and CSTCPP; Birman{ Ko{Lee monoid; Kronecker elliptic sigma functions 2010 Mathematics Subject Classification: 14N15; 53D45; 16W30 To the memory of Alain Lascoux 1944{2013, the great Mathematician, from whom I have learned a lot about the Schubert and Grothendieck polynomials. arXiv:1502.00426v3 [math.RT] 5 Jan 2016 Contents 1 Introduction 6 2 Dunkl elements 18 2.1 Some representations of the algebra 6DTn ........................... 19 2.1.1 Dynamical Dunkl elements and equivariant quantum cohomology . 19 2.1.2 Step functions and the Dunkl{Uglov representations of the degenerate affine Hecke algebras [138]....................................... 25 2.1.3 Extended Kohno{Drinfeld algebra and Yangian Dunkl{Gaudin elements . 26 2.2 \Compatible" Dunkl elements, Manin matrices and algebras related with weighted complete graphs rKn ....................................... 27 2.3 Miscellany . 29 2.3.1 Non-unitary dynamical classical Yang{Baxter algebra DCYBn ........... 29 2.3.2 Dunkl and Knizhnik{Zamolodchikov elements . 31 2 A.N. Kirillov 2.3.3 Dunkl and Gaudin operators . 32 2.3.4 Representation of the algebra 3Tn on the free algebra Zht1; : : : ; tni ......... 33 2.3.5 Kernel of Bruhat representation . 34 2.3.6 The Fulton universal ring [47], multiparameter quantum cohomology of flag varieties [45] and the full Kostant{Toda lattice [29, 80]................ 36 3 Algebra 3HTn 38 3.1 Modified three term relations algebra 3MTn(β; )...................... 40 3.1.1 Equivariant modified three term relations algebra . 42 3.2 Multiplicative Dunkl elements . 44 3.3 Truncated Gaudin operators . 46 3.4 Shifted Dunkl elements di and Di ................................ 49 (0) 4 Algebra 3Tn (Γ) and Tutte polynomial of graphs 52 4.1 Graph and nil-graph subalgebras, and partial flag varieties . 52 (0) 4.1.1 Nil-Coxeter and affine nil-Coxeter subalgebras in 3Tn ............... 52 4.1.2 Parabolic 3-term relations algebras and partial flag varieties . 54 4.1.3 Universal Tutte polynomials . 62 4.1.4 Quasi-classical and associative classical Yang{Baxter algebras of type Bn ..... 69 4.2 Super analogue of 6-term relations and classical Yang{Baxter algebras . 71 ! 4.2.1 Six term relations algebra 6Tn, its quadratic dual (6Tn) , and algebra 6HTn .... 71 (0) F 4.2.2 Algebras 6Tn and 6Tn ................................. 73 4.2.3 Hilbert series of algebras CYBn and 6Tn ........................ 76 4.2.4 Super analogue of 6-term relations algebra . 79 4.3 Four term relations algebras / Kohno{Drinfeld algebras . 80 4.3.1 Kohno{Drinfeld algebra 4Tn and that CYBn ..................... 80 + 4.3.2 Nonsymmetric Kohno{Drinfeld algebra 4NTn, and McCool algebras PΣn and PΣn 83 4.3.3 Algebras 4TTn and 4STn ................................ 85 0 4.4 Subalgebra generated by Jucys{Murphy elements in 4Tn ................... 86 4.5 Nonlocal Kohno{Drinfeld algebra NL4Tn ........................... 87 4.5.1 On relations among JM-elements in Hecke algebras . 89 4.6 Extended nil-three term relations algebra and DAHA, cf. [24]................ 90 4.7 Braid, affine braid and virtual braid groups . 95 4.7.1 Yang{Baxter groups . 97 4.7.2 Some properties of braid and Yang{Baxter groups . 97 4.7.3 Artin and Birman{Ko{Lee monoids . 99 5 Combinatorics of associative Yang{Baxter algebras 101 5.1 Combinatorics of Coxeter element . 102 5.1.1 Multiparameter deformation of Catalan, Narayana and Schr¨odernumbers . 109 5.2 Grothendieck and q-Schr¨oderpolynomials . 110 5.2.1 Schr¨oderpaths and polynomials . 110 5.2.2 Grothendieck polynomials and k-dissections . 114 5.2.3 Grothendieck polynomials and q-Schr¨oderpolynomials . 115 5.2.4 Specialization of Schubert polynomials . 120 5.2.5 Specialization of Grothendieck polynomials . 133 5.3 The \longest element" and Chan{Robbins{Yuen polytope . 134 5.3.1 The Chan{Robbins{Yuen polytope CRYn ....................... 134 5.3.2 The Chan{Robbins{M´esz´arospolytope Pn;m ..................... 139 5.4 Reduced polynomials of certain monomials . 143 5.4.1 Reduced polynomials, Motzkin and Riordan numbers . 147 5.4.2 Reduced polynomials, dissections and Lagrange inversion formula . 149 On Some Quadratic Algebras 3 A Appendixes 153 A.1 Grothendieck polynomials . 153 A.2 Cohomology of partial flag varieties . 155 A.3 Multiparamater 3-term relations algebras . 159 A.3.1 Equivariant multiparameter 3-term relations algebras . 159 A.3.2 Algebra 3QTn(β; h), generalized unitary case . 161 A.4 Koszul dual of quadratic algebras and Betti numbers . 162 0 A.5 On relations in the algebra Zn ................................. 163 0 0! A.5.1 Hilbert series Hilb 3Tn ; t and Hilb 3Tn ; t : Examples . 165 A.6 Summation and Duality transformation formulas [63]..................... 166 References 167 Extended abstract We introduce and study a certain class of quadratic algebras, which are nonhomogeneous in general, together with the distinguish set of mutually commuting elements inside of each, the so-called Dunkl elements. We describe relations among the Dunkl elements in the case of a family of quadratic algebras corresponding to a certain splitting of the universal classical Yang{Baxter relations into two three term relations. This result is a further extension and generalization of analogous results obtained in [45, 117] and [76]. As an application we describe explicitly the set of relations among the Gaudin elements in the group ring of the symmetric group, cf. [108]. We also study relations among the Dunkl elements in the case of (nonhomogeneous) quadratic algebras related with the universal dynamical classical Yang{Baxter relations. Some relations of results obtained in papers [45, 72, 75] with those obtained in [54] are pointed out. We also identify a subalgebra generated by the generators corresponding to the simple roots in the extended Fomin{Kirillov algebra with the DAHA, see Section 4.3. The set of generators of algebras in question, naturally corresponds to the set of edges of the complete graph Kn (to the set of edges and loops of the complete graph with (simple) loops Ken in dynamical and equivariant cases). More generally, starting from any subgraph Γ of the complete (0) graph with simple loops Ken we define a (graded) subalgebra 3Tn (Γ) of the (graded) algebra (0) 3Tn (Ken)[70]. In the case of loop-less graphs Γ ⊂ Kn we state conjecture, Conjecture 4.15 (0) ab in the main text, which relates the Hilbert polynomial of the abelian quotient 3Tn (Γ) of the (0) 12 algebra 3Tn (Γ) and the chromatic polynomial of the graph Γ we are started with . We check 1We expect that a similar conjecture is true for any finite (oriented) matroid M. Namely, one (A.K.) can define an analogue of the three term relations algebra 3T (0)(M) for any (oriented) matroid M. We expect that the abelian quotient 3T (0)(M)ab of the algebra 3T (0)(M) is isomorphic to the Orlik{Terao algebra [114], denoted by OT(M) (known also as even version of the Orlik{Solomon algebra, denoted by OS+(M) ) associated with matroid M [28]. Moreover, the anticommutative quotient of the odd version of the algebra 3T (0)(M), as we expect, is isomorphic to the Orlik{Solomon algebra OS(M) associated with matroid M, see, e.g., [11, 49]. In particular, Hilb(3T (0)M)ab; t = tr(M)TutteM; 1 + t−1; 0: We expect that the Tutte polynomial of a matroid, Tutte(M; x; y), is related with the Betti polynomial of a matroid 2 (0) 2 M. Replacing relations uij = 0, 8 i; j, in the definition of the algebra 3T (Γ) by relations uij = qij , 8 i; j, (i; j) 2 E(Γ), where fqij g(i;j)2E(Γ), qij = qji, is a collection of central elements, give rise to a quantization of the Orlik{Terao algebra OT(Γ). It seems an interesting task to clarify combinatorial/geometric significance of noncommutative versions of Orlik{Terao algebras (as well as Orlik{Solomon ones) defined as follows: OT (Γ) := 3T (0)(Γ), its \quantization" 3T (q)(Γ)ab and K-theoretic analogue 3T (q)(Γ; β)ab, cf. Definition 3.1, in the theory of hyperplane arrangements. Note that a small modification of arguments in [89] as were used for the proof of our ab Conjecture 4.15, gives rise to a theorem that the algebra 3Tn(Γ) is isomorphic to the Orlik{Terao algebra OT(Γ) studied in [126].