Browsing by Author "Smith, Larry"

We report the discovery of 76 new T dwarfs from the UKIRT Infrared Deep Sky Survey (UKIDSS) Large Area Survey (LAS). Near-infrared broad- and narrow-band photometry and spectroscopy are presented for the new objects, along with Wide-field Infrared Survey Explorer (WISE) and warm-Spitzer photometry. Proper motions for 128 UKIDSS T dwarfs are presented from a new two epoch LAS proper motion catalogue. We use these motions to identify two new benchmark systems: LHS 6176AB, a T8p+M4 pair and HD 118865AB, a T5.5+F8 pair. Using age constraints from the primaries and evolutionary models to constrain the radii, we have estimated their physical properties from their bolometric luminosity. We compare the colours and properties of known benchmark T dwarfs to the latest model atmospheres and draw two principal conclusions. First, it appears that the H - [4.5] and J - W2 colours are more sensitive to metallicity than has previously been recognized, such that differences in metallicity may dominate over differences in T-eff when considering relative properties of cool objects using these colours. Secondly, the previously noted apparent dominance of young objects in the late-T dwarf sample is no longer apparent when using the new model grids and the expanded sample of late-T dwarfs and benchmarks. This is supported by the apparently similar distribution of late-T dwarfs and earlier type T dwarfs on reduced proper motion diagrams that we present. Finally, we present updated space densities for the late-T dwarfs, and compare our values to simulation predictions and those from WISE.

Let rho : G --> GL(n, F) be a representation of a finite group over the field F, V = F-n the corresponding G-module, and F[V] the algebra of polynomial functions on V. The action of G on V extends to F[ V], and F[V](G), respectively F[V](G), denotes the ring of invariants, respectively coinvariants. The theorem of Steinberg referred to in the title says that when F = C, dim(C) (Tot(C[V](G))) = \G\ if and only if G is a complex reflection group. Here Tot(F[V](G)) denotes the direct sum of all the homogeneous components of the graded algebra F[V](G) and \G\ is the order of G. Chevalley's theorem tells us that the ring of invariants of a complex pseudoreflection representation G --> GL(n, C) is polynomial algebra, and the theorem of Shephard and Todd yields the converse. Combining these results gives: dim(F)(Tot(C[V](G)) = \G\ if and only if C[V](G) is a polynomial algebra. The purpose of this note is to show that the two conditions (i) dim(F)(Tot(F[V](G))) = \G\, (ii) F[V](G) is a polynomial algebra are equivalent regardless of the ground field; in particular in the modular case.

Let F-q be the finite field with q elements, q = p(v), p is an element of N a prime, and Mat(2.2)(F-q) the vector space of 2 x 2-matrices over F. The group GL(2, F) acts on Mat(2,2)(F-q) by conjugation. In this note, we determine the invariants of this action. In contrast to the case of an infinite field, where the trace and determinant generate the ring of invariants, several new invariants appear in the case of finite fields. (C) 2002 Elsevier Science (USA).

Let rho : G hooked right arrow GL (n, F) be a representation of a finite group G over the field F, and denote by V the vector space F(n) on which G acts via rho. By means of the dual (contragredient) representation G also acts on the symmetric algebra S(V ) of the vector space V dual to V. Following [10] we denote S(V ) by F[V] and regard it as the algebra of polynomial functions(1) on V. The subalgebra of polynomials invariant under this action is denoted by F[V](G). If U subset of V = F(n) is a linear subspace then the pointwise stabilizer of U is denoted by G(U) = {g is an element of G vertical bar g(u) = u for all u is an element of U}. It is known that several properties of F[V](G) are inherited by F[V](GU) (see, e. g., [6] Section 10.6 and the references there). For finite fields, following the pioneering work of Dwyer and Wilkerson [1], many such properties have been demonstrated using the T-functor introduced by Lannes [4] ( see also [9]) in his study of unstable modules over the Steenrod algebra. In this note we show that given a degree bound for the generators of F[V](G) as an algebra, this bound is inherited by F[V](GU) when F = F(q) is a Galois field with q elements. To do so we examine some finiteness properties of unstable algebras over the Steenrod algebra and show that the T-functor preserves them, extending results in [6] Section 10.2.

Let rho : G --> GL(n,F) be a representation of a finite group G over the field IF. The group G acts on the algebra of polynomial functions F[V] on V via rho and the subalgebra of polynomials invariant under this action is denoted by F[V](G). If U subset of or equal to V = F-n is a linear subspace then the pointwise stabilizer of U is denoted by G(U) = {g is an element of G \g(u) = u For All u is an element of U}. In this note we examine the relation between F[V](G) and F[V](Gu) when F = F-q is a Galois field with q elements using the T-functor introduced by J. Lannes [13]. We show that a wide variety of properties of F[V](G) are inherited by F[V](Gu). For example, among other things: (1) we reprove a result of R. Steinberg [26] and H. Nakajima [17] that F[V](Gu) is a polynomial algebra when IF[V]G is; (2) we show that the Cohen-Macaulay property is inherited by F[V](Gu) from F[V](G); (3) and when F[V](G) is a complete intersection, then so is F[V](Gu). We apply the T-functor to study degree bounds for generators of rings of invariants, show how the T-functor relates to the transfer homomorphism Tr-G:F[V] --> F[V](G), and give an application in the modular case.

Let rho : G --> GL(n, F) be a faithful representation of the finite group G over the held F. In 1916, E. Noether proved that for F of characteristic zero the ring of invariants F[V](G) is generated as an algebra by the invariant polynomials of degree at most \G\. This result has been generalized to the case where the characteristic of F is greater than \G\, or when the characteristic of F is prime to the order of G and the group G is solvable. In this note we show how to refine Noether's proof to yield a more general nonmodular result. In particular we prove that Noether's bound holds for the alternating groups in the nonmodular case.

Let G be a finite group and rho: G --> GL(n; C) a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions C[V](G) on the vector space V = C-n is generated by homogeneous polynomials of degree at most beta, where beta is the largest degree of a generator in a minimal generating set for C[reg(C)(G)](G), and reg(C)(G) is the complex regular representation of G. In this note we give a new proof of this result, and at the same time extend it to fields F whose characteristic p is larger than \G\, the order of the group G.

Let rho : G curved right arrow GL(n,F) be a representation of a finite group G over the field F. Denote by F[V] the algebra of polynomial functions on the vector space V = F-n. The group G acts on V and hence also on F[V]. The algebra of coinvariants is F[V](G) = F[V]/h(G), where h(G) subset of F[V] is the ideal generated by all the homogeneous G-invariant forms of strictly positive degree. If the field F has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that F[V](G) is a Poincare duality algebra if and only if G is a pseudoreflection group. In this note we explore the situation for fields of nonzero characteristic. We prove an analogue of Steinberg's theorem for the case n = 2 and give a counterexample in the modular case when n = 4.

In a paper on F-rationality [J. Algebra 176 (1995) 824-860] Donna Glassbrenner showed that over a field of odd characteristic p the Hilbert ideals of the tautological representations of the symmetric group Sigma(n) and alternating group A(n) coincide if n equivalent to 0, 1 mod p. She asked if this was always the situation in the modular case. We answer this in the affirmative using Macaulay's theory of irreducible ideals in polynomial algebras: a somewhat forgotten bywater of commutative algebra. As a bonus, the method yields applications back to the original question of F-rationality studied in [J. Algebra 176 (1995) 824-860]. (C) 2004 Elsevier Inc. All rights reserved.

Let f(Z) is an element of F[Z] be a univariate, separable polynomial of degree n with roots x(1), ..., x(n) in some algebraic closure F of the ground field F. It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on F(n) by permutation of the coordinates and each P is an element of F[X(1), ..., X(n)](G) defines a relation P-P(x(1), ..., x(n)) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an H-basis of G-invariant relations if an donly if the algebra of coinvariants F[X(1), ..., X(n)](G) has dimension vertical bar G vertical bar over F. To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite group G has dimension vertical bar G vertical bar if and only if G = Sigma(n) acting via the tautological permutation representation.

Let d(2,0) = x(2)y + xy(2), d(2,1) = x(2) + xy + y(2) is an element of F-2[x,y] be the two Dickson polynomials. If a and b are positive integers, the ideal (d(2,0)(a), d(2,1)(b)) subset of F-2[x, y] is invariant under the action of the mod 2 Steenrod algebra A if and only if when we write b = 2(t) . k with k odd, then a less than or equal to 2(t). The quotient algebra F-2[x, y]/(d(2,0)(a), d(2,1)(b)) is a Poincare duality algebra and for such a and b admits an unstable action of A . It has trivial Wu classes if and only if a = 2(t) for some t greater than or equal to 0 and b = 2(t)(2(s) - 1) for some s > 0. We ask under what conditions on a and b, F-2[x, y]/(d(2,0)(a), d(2,1)(b)) appears as the mod 2 cohomology of a manifold. In this note we show that for a = 2(t) = b there is a topological space whose cohomology is F-2[x,y]/(d(2,0)(2t), d(2,1)(2t)) if and only if t = 0, 1, 2, or 3, and in these cases the space may be taken to be a smooth manifold.

We show that a P- submodule of an unstable Noetherian R-module over a Noetherian ring R has a P- -invariant primary decomposition.