In the Coxeter group W (R) generated by the root system R, let T (R) be the number of conjugacy classes having no eigenvalue +1 and let S(R) be the number of conjugacy classes having no eigenvalue −1. The algebra H W (R) of observables of the rational Calogero model based on the root system R possesses T (R) independent traces; the same algebra, considered as an associative superalgebra with respect to a certain natural parity, possesses S(R) even independent supertraces and no odd trace or supertrace. The numbers T (R) and S(R) are determined for all irreducible root systems (hence for all root systems). It is shown that T (R) ≤ S(R), and T (R) = S(R) if and only if superalgebra H W (R) contains a Klein operator (or, equivalently, W (R) ∋ −1).Keywords: Trace; supertrace; Cherednik algebra; algebra of observables; Calogero model. Mathematics Subject Classification: 17B80, 16W551. Definitions and generalities TracesLet A be an associative superalgebra with parity π. All expressions of linear algebra are given for homogeneous elements only and are supposed to be extended to inhomogeneous elements via linearity.A linear function str on A is called a supertrace ifA linear function tr on A is called a trace if S.E. Konstein and R. StekolshchikLet A 1 and A 2 be associative superalgebras with parities π 1 and π 2 , respectively. Define the tensor product A = A 1 ⊗A 2 as a superalgebra with the product (a 1 ⊗a 2 )(b 1 ⊗b 2 ) = (a 1 b 1 )⊗(a 2 b 2 ) (no sign factors in this formula) and the parity π defined by the formula π(a ⊗ b) = π 1 (a) + π 2 (b).Let T i be a trace on A i . Clearly, the function T such thatis an even supertrace on A . Klein operatorLet A be an associative superalgebra with parity π. Following M.Vasiliev, see, e.g.[21], we say that an element K ∈ A is a Klein operator a if π(K) = 0, K f = (−1) π( f ) f K for any f ∈ A and K 2 = 1. Every Klein operator belongs to the anticenter of the superalgebra A , see [18], p.41. b Any Klein operator, if exists, establishes an isomorphism between the space of even traces and the space of even supertraces on A . Namely, if f → T ( f ) is an even trace, then f → T ( f K) is a supertrace, and if f → S( f ) is an even supertrace, then f → S( f K) is a trace. Group algebraLet V = R n and G ⊂ End(V ) be a finite group. The group algebra C[G] of G consists of all linear combinations ∑ g∈W (R) α gḡ , where α g ∈ C. We distinguish g considered as an element of the group G ⊂ End(V ) from the same elementḡ ∈ C[G] considered as an element of the group algebra. The addition in C[G] is defined as follows:and the multiplication is defined by setting g 1 g 2 = g 1 g 2 .Note that the additions in C[G] and in End(V ) differ. For example, if I ∈ G is unity and the matrix J = −I from End(V ) belongs to G, then I + J = 0 in End(V ) while I + J = 0 in C[G]. Root systemsLet V = R N be endowed with a non-degenerate symmetric bilinear form (·, ·) and the vectors a i constitute an orthonormal basis in V , i.e. Co-published by Atlantis Press and Taylor & Francis Copyright: the authors 29...

No abstract

The notions of a perfect element and an admissible element of the free modular lattice D r generated by r ≥ 1 elements are introduced by Gelfand and Ponomarev in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii , IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3-58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1-56]. We recall that an element a ∈ D of a modular lattice L is perfect, if for each finite-dimension indecomposable K -linear representation ρ X : L → L(X ) over any field K , the image ρ X (a) ⊆ X of a is either zero, or ρ X (a) = X , where L(X ) is the lattice of all vector K -subspaces of X . A complete classification of such elements in the lattice D 4 , associated to the extended Dynkin diagram D 4 (and also in D r , where r > 4) is given in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii (1901-1973), IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3-58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1-56; I.M. Gelfand, V.A. Ponomarev, Lattices, representations, and their related algebras, I, Uspehi Mat. Nauk 31 (5(191)) (1976) 71-88 (Russian); English translation: Russian Math. Surv. 31 (5) (1976) 67-85; I.M. Gelfand, V.A. Ponomarev, Lattices, representations, and their related algebras, II. Uspehi Mat. Nauk 32 (1(193)) (1977) 85-106 (Russian); English translation: Russian Math. Surv. 32 (1) (1977) 91-114]. The main aim of the present paper is to classify all the admissible elements and all the perfect elements in the Dedekind lattice D 2,2,2 generated by six elements that are associated to the extended Dynkin diagram E 6 . We recall that in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii , IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3-58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1-56], Gelfand and Ponomarev construct admissible elements of the lattice D r recurrently. We suggest a direct method for creating admissible elements. Using this method we also construct admissible elements for D 4 and show that these elements coincide modulo linear equivalence with admissible elements constructed by Gelfand and Ponomarev. Admissible sequences and admissible elements for D 2,2,2 (resp. D 4 ) form 14 classes (resp. 8 classes) and possess some periodicity.Our classification of perfect elements for D 2,2,2 is based on the description of admissible elements. The constructed set H + of perfect elements is the union of 64-element distributive lattices H + (n), and H + is the distributive lattice itself. The lattice of perfect elements B + obtained by Gelfand and Ponomarev for D 4 can be imbedded into the lattice of perfect elements H + , associated with D 2,2,2 . 96 R. Stekolshchik / Journal of Pure and Applied Algebra 211 (2007) 95-202 Herrmann in...

No abstract

In 1972, R. Carter introduced admissible diagrams to classify conjugacy classes in a finite Weyl group W . For any two non-orthogonal roots α and β corresponding to vertices of admissible diagram, we draw the dotted (resp. solid) edge {α, β} if (α, β) > 0 (resp. (α, β) < 0). The diagram with properties of admissible diagram and possibly containing dotted edges are said to be Carter diagrams. For any Carter diagram Γ, we introduce the partial Cartan matrix BΓ, which is analogous to the Cartan matrix associated with a Dynkin diagram. A linkage diagram is obtained from Γ by adding an extra root γ, together with its bonds, so that the resulting subset of roots is linearly independent. With every linkage diagram we associate the linkage label vector γ ∇ , similar to the "numerical labels" introduced by Dynkin for the study of irreducible linear representations of the semisimple Lie algebras. The linkage diagrams connected under the action of dual partial Weyl group W ∨ S (associated with BΓ) constitute the set L (Γ) dubbed the linkage system. For any simply-laced Carter diagram Γ, the linkage system L (Γ) is explicitly constructed. To obtain linkage diagrams θ ∇ ∈ L (Γ), we use an easily verifiable criterion: B ∨ Γ (θ ∇ ) < 2, where B ∨ Γ is the quadratic form associated with B −1 Γ . A Dynkin diagram Γ ′ such that rank(Γ ′ ) = rank(Γ) + 1 and any Γ-associated root subset S lies in the root system Φ(Γ ′ ), will be called the Dynkin extension of the Carter diagram Γ and denoted by Γ