No abstract

ABSTRACT. Let W be a Weyl group and let W' be a parabolic subgroup of W . Define A as follows: ®Q[uj .9f(W) where .9f (W) is the generic algebra of type An over Q [u] , u an indeterminate, associated with the group W, and R is a Q[u]-algebra, possibly of infinite rank, in which u is invertible. Similarly, we define A' associated with W'. Let M be an A-A bimodule, and let b EM. Define the relative normwhere T is the set of distinguished right coset representives for W' in W.In addition, other properties of the relative norm are given and used to develop a theory of induced modules for generic Heeke algebras including a Markey decomposition. This section of the paper is previously unpublished work of P. Hoefsmit and L. L. Scott. with 'W; a k;-cycle of length k; -I in Ski' Then the main result of this paper IS Theorem. The set {b" I rd-n} is a basis for ZA(Sn) (A(Sn)) over Q [u, u-1 ] .Remark. The norms b" in ZA(Sn)(A(Sn)) are analogs of conjugacy class sums in the center of QSn and, in fact, specialization of these norms at u = I gives the standard conjugacy class sum basis of the center of QSn up to coefficients from Q.

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

Let [Formula: see text] with [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote, respectively, the symmetric group and alternating group on [Formula: see text] letters. Let [Formula: see text] be an indeterminate, and define [Formula: see text] where [Formula: see text] are certain prescribed forms in [Formula: see text]. For a certain set of these forms, we show unconditionally that there exist infinitely many primes [Formula: see text] such that [Formula: see text] is irreducible over [Formula: see text], [Formula: see text], and the fields [Formula: see text] are distinct and monogenic, where [Formula: see text]. Using a different set of forms, we establish a similar result for all square-free values of [Formula: see text], with [Formula: see text], and any positive integer value of [Formula: see text] for which [Formula: see text] is square-free. Additionally, in this case, we prove that [Formula: see text]. Finally, we show that these results can be extended under the assumption of the [Formula: see text]-conjecture. Our methods make use of recent results of Helfgott and Pasten.

ABSTRACT. Let W be a Weyl group and let W' be a parabolic subgroup of W . Define A as follows: ®Q[uj .9f(W) where .9f (W) is the generic algebra of type An over Q [u] , u an indeterminate, associated with the group W, and R is a Q[u]-algebra, possibly of infinite rank, in which u is invertible. Similarly, we define A' associated with W'. Let M be an A-A bimodule, and let b EM. Define the relative normwhere T is the set of distinguished right coset representives for W' in W.In addition, other properties of the relative norm are given and used to develop a theory of induced modules for generic Heeke algebras including a Markey decomposition. This section of the paper is previously unpublished work of P. Hoefsmit and L. L. Scott. with 'W; a k;-cycle of length k; -I in Ski' Then the main result of this paper IS Theorem. The set {b" I rd-n} is a basis for ZA(Sn) (A(Sn)) over Q [u, u-1 ] .Remark. The norms b" in ZA(Sn)(A(Sn)) are analogs of conjugacy class sums in the center of QSn and, in fact, specialization of these norms at u = I gives the standard conjugacy class sum basis of the center of QSn up to coefficients from Q.