A generalization of Eichler's trace formula

We introduce Brauer algebras associated to complex reflection groups of type G(m, p, n), and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic zero.

Linear Algebra and its Applications, 1996

Let ti be a central simple algebra over a field k, and let tr denote its reduced trace. Then qd(X) = tr X" is the trace form of degree d on &. A similarity of W, Vd) is a linear map f : d +d for which there exists A E k" such that ~d(f(U>) = **da> for all a Ed. The "standard" similarities are the ones obtained as compositions of automorphisms. antiautomorphisms, and scalar maps of the algebra S? THEOREM. Suppose d > 2 and d! is nonzero in k. Then (&, 'PC{> is regular indecomposable, and every similarity is standard.

Bulletin of the American Mathematical Society

Communicated by Nathan Jacobson, December 1, 1969 1. Introduction. Let F be a quadratic vector space over the field of rational numbers Q. We assume that the associated quadratic form q is positive definite with square discriminant. Let M be a lattice in V which is maximal integral with respect to q. We denote by H the number of proper classes of maximal integral lattices. The purpose of this note is to announce a formula for H. This formula is derived by applying the Selberg Trace Formula in an appropriate manner. The method we employ is motivated by the successful use of the Selberg Trace Formula in the computation of ideal class numbers of quaternion algebras over Q (cf. [ó]). Since q has square discriminant, we may assume that V = 21, a quaternion (division) algebra over Q, and q = N, the norm form of 31. We may take M to be 0, a fixed maximal order in §1. If a basis of 0 over the ring of integers Z, then the discriminant of 0 with respect to the norm form N is = | Tr(#»•#ƒ) |-D, the discriminant of the quaternion algebra 2Ï. Here * is the canonical involution of 21. It is well known that D = d 2 , where d is a positive square-free integer. Let us write d-p\ • • • p e , where the pi, i = l, • • • , e are distinct prime numbers. We recall that {pu • • • , p e \ is the set of finite primes p such that % P = %®QQ P is a division algebra over Q p , the field of £-adic numbers. One calls pi, • • • , p e the nonsplit or ramified primes of 2Ï. We do not apply the Selberg Trace Formula in the setting afforded by the orthogonal groups which appear in the usual definition of H. The reason is that the definitions of these groups involve a norm condition which makes integration unmanageable and which also complicates conjugacy considerations. To avoid these difficulties, we replace the usual definition of H by one which is more suitable for our

Annales des Sciences Mathematiques du Quebec

Let p be a prime and suppose that K/F is a cyclic extension of degree p^n with group G. Let J be the F_pG-module K^*/K^{*p} of pth-power classes. In our previous paper we established precise conditions for J to contain an indecomposable direct summand of dimension not a power of p. At most one such summand exists, and its dimension must be p^i+1 for some 0<=i<n. We show that for all primes p and all 0<=i<n, there exists a field extension K/F with a summand of dimension p^i+1.

Journal of Algebra, 1989

Bulletin of the London Mathematical Society

We study the trace form qL of G-Galois algebras L/K when G is a finite group and K is a field of characteristic different from 2. We introduce in this paper the category of 2-reduced groups and, when G is such a group, we use a formula of Serre to compute the second Hasse-Witt invariant of qL. By combining this computation with work of Quillen we determine the isometry class of qL for large families of G-Galois algebras over global fields. We also indicate how our results generalize to Galois G-covers of schemes.

1999

The Iwahori-Hecke algebra has a canonicaltrace $\tau$. The trace is the evaluation at the identity element in the usual interpretation of the Iwahori-Hecke algebra as a sub-algebra of the convolution algebra of a p-adic semi-simple group. The Iwahori-Hecke algebra contains an important commutative sub-algebra ${\bf C}[\theta_x]$, that was described and studied by Bernstein, Zelevinski and Lusztig. In this note we compute the generating function for the value of $\tau$ on the basis $\theta_x$.

Communications in Algebra, 2009

We provide a sufficient condition that ensures the nilpotency of endomorphisms universally of trace zero of Schur-finite objects in a category of homological type, i.e., a Q-linear ⊗-category with a tensor functor to super vector spaces. This generalizes previous results about finite-dimensional objects, in particular by Kimura in the category of motives. We also present some facts which suggest that this might be the best generalization possible of this line of proof. To get the result we prove an identity of trace relations on super vector spaces which has an independent interest in the field of combinatorics. Our main tool is Berele-Regev's theory of Hook Schur functions. We use their generalization of the classic Schur-Weyl duality to the "super" case, together with their factorization formula.

Pacific Journal of Mathematics, 1991

Every algebra p.i. equivalent to some M k / and with zero annihilator of the Razmyslov ideal has a central extension with non-degenerate trace. A p.i. algebra R is said to be verbally prime if whenever f(x\, ... , x n)g(x n +ι, ... , x m) is an identity for R then either / or g is. Kemer introduced these algebras in [3]. Since then, verbally prime algebras have been the subject of a number of papers. At the end of this paper we will summarize the results known to us. A common theme is that verbally prime algebras have many properties in common with prime p.i. algebras. In the present work we continue this thread and show that certain verbally prime algebras have "nice" embeddings into trace rings. Kemer proved that every non-trivial verbally prime p.i. algebra in characteristic zero must be p.i. equivalent to either: n x n matrices over the field; or n x n matrices over an infinite dimensional Grassmann algebra E or M k j. The algebra M k j is a certain subalgebra of the (k+l) x(fc+/)-matrices over E. E has a natural Z/2Z-grading E = EQ Θ E\, in which EQ is spanned by the even words and E\ is spanned by the odd words. Then M k j consists of all (k + l) x (k + /) matrices of the form (£ £) > where A is a k x fc-matrix with entries in EQ, D is an / x /-matrix with entries EQ , and B and C have entries in E\. The algebra M k j has a trace function Xr.M^j-» EQ defined by tΓ ((c ^)) = tr ^4-tr Z). This function satisfies all of the usual properties of the usual properties of trace: it takes values in EQ , the center of M k j it is iso-linear; and tr(xy) = \τ(yx) for all x, y e M k j. Razmyslov [5] studied the central polynomials and trace identities of M k j. He found non-vanishing, multilinear central polynomials p(x\ 9 ... , x n , a) and c(x\ 9 ... , x n) for M k j with the property that (*) p(x\, ... , x n , a) = c(xχ, ... , Xn)ΐr(a) on M k j. For convenience, we generally abbreviate p(x\, ... , x n , a) to p(x, a) and c(x\, ... 9 x n) to c(x).

Pacific Journal of Mathematics, 1970

Some time ago Clifford described the behavior of an irreducible representation of a finite group when it is restricted to a normal subgroup. One interesting case in this description requires that the representation be written in an algebraically closed field. In this note we shall consider this case when the field is "small". We describe conditions under which an irreducible representation decomposes as the tensor product of two projective representations. Our approach uses certain subalgebras of the group algebra and the course of the discussion makes it fairly easy to keep track of the division algebras that appear. Hence we obtain some information about the Schur index. We apply this information to the case where the group is a semi-direct product PA of a p-group P and a normal cyclic group A. If J^"~ is an algebraic number field and χ an absolutely irreducible character of PA, then there normal subgroups Pi 2 P2 Ξ> P 3 of P which contain C P (A) such that the Schur index m> (χ) of χ over J^* divides 2[P λ \ P 2 ]e where e is the exponent of P2/P3. The factor 2 can be omitted if p Φ 2. Some conditions are available to restrict the Pi further. 1* Preliminaries* In this section we summarize the results about the Schur index and Clifford's theory that will be used later. Let G be a finite group, j^ a field of characteristic zero, M an irreducible ^'(G)-module with character θ.