Bases for rings of coinvariants

We study the multiplicative structure of rings of coinvariants for finite groups. We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants. We apply our methods to the Dickson, upper triangular and symmetric coinvariants. Along the way, we recover theorems of Steinberg [17] and E. Artin [1]. Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada     

Rings of invariants for modular representations of elementary abelian <Emphasis Type="Italic">p</Emphasis>-groups

We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two-dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two-dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three-dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three-dimensional representation of an elementary abelian p-group is a complete intersection.     

Chern–Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

Depth of modular invariant rings

It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be Cohen-Macaulay and computing the depth is often very difficult. In this paper¹ we obtain a simple formula for the depth of the ring of invariants for a family of modular representations. This family includes all modular representations of cyclic groups. In particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [6].     

A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants

A special case of Haiman?s identity [M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, Invent. Math. 149 (2002) 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in q,t. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman?s formula for the Hilbert series into an explicit polynomial in q,t with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.     

Gröbner bases for the Hilbert ideal and coinvariants of the dihedral group D2p

We consider a finite dimensional representation of the dihedral group D_2p over a field of characteristic two where p is an odd integer and study the corresponding Hilbert ideal I_H. We show that I_H has a universal Gröbner basis consisting of invariants and monomials only. We provide sharp bounds for the degree of an element in this basis and in a minimal generating set for I_H. We also compute the top degree of coinvariants when p is prime.     

Regularity of lex-segment ideals: Some closed formulas and applications

A closed formula for computing the regularity of the lex-segment ideal in terms of the Hilbert function is given. This regularity bounds the one of any ideal with the same Hilbert function. As a consequence, we give explicit expressions to bound the regularity of a projective scheme in terms of the coefficients of the Hilbert polynomial.

We also characterize, in terms of their coefficients, which polynomials are Hilbert polynomials of some projective scheme.

Finally, we provide some applications to estimates for the maximal degree of generators of Gröbner bases in terms of the degrees of defining equations.

     

The relative trace ideal and the depth of modular rings of invariants

We prove that for a modular representation, the depth of the ring of invariants is the sum of the dimension of the fixed point space of the p-Sylow subgroup and the grade of the relative trace ideal. We also determine which of the Dickson invariants lie in the radical of the relative trace ideal and we describe how to use the Dickson invariants to compute the grade of the relative trace ideal.     

Generalized multiplicities of edge ideals

We explore connections between the generalized multiplicities of square-free monomial ideals and the combinatorial structure of the underlying hypergraphs using methods of commutative algebra and polyhedral geometry. For instance, we show that the j-multiplicity is multiplicative over the connected components of a hypergraph, and we explicitly relate the j-multiplicity of the edge ideal of a properly connected uniform hypergraph to the Hilbert–Samuel multiplicity of its special fiber ring. In addition, we provide general bounds for the generalized multiplicities of the edge ideals and compute these invariants for classes of uniform hypergraphs.     

Nonnegative unitary operators

Unitary operators in Hilbert space map an orthonormal basis onto another. In this paper we study those that map an orthonormal basis onto itself. We show that a sequence of cardinal numbers is a complete set of unitary invariants for such an operator. We obtain a characterization of these operators in terms of their spectral properties. We show how much simpler the structure is in finite-dimensional space, and also describe the structure of certain isometries in Hilbert space.

     

The Hochschild cohomology of square-free monomial complete intersections

We determine the Hochschild cohomology algebras of the square-free monomial complete intersections. In particular we provide an explicit formula for the cup product which gives the cohomology module an algebra structure and then we describe this structure in terms of generators and relations. In addition, we compute the Hilbert series of the Hochschild cohomology of these algebras.     

Trees, parking functions, syzygies, and deformations of monomial ideals

For a graph , we construct two algebras whose dimensions are both equal to the number of spanning trees of . One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to -parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

     

Harmonic Analysis of Fractal Processes via C* - Algebras

We construct a harmonic analysis of iteration systems which include those which arise from wavelet algorithms based on multiresolutions. While traditional discretizations lead to asymptotic formulas, we argue here for a direct Fourier duality; but it is based on a non - commutative harmonic analysis, specifically on representations of the Cuntz C* -algebras. With this approach the waling from the wavelet takes the form of an endomorphism of B(H), H a Hilbert space derived from the lattice of translations. We use this to describe, and to calculate, new invariants for the wavelets. those iteration systems which arise from wavelets and from Julia sets, we show that the associated endomorphisms are in fact Powers shifts.     

On the Height of the Minimal Hilbert Basis

We present an elementary proof of a result due to Ewald and Wessels: in a pointed, polyhedral cone of dimension n ≥ 3 with integer-valued generators, any linearly independent generator representation for a minimal Hilbert basis element has coefficient sum less than n — 1. Our proof makes explicit use of the geometry of the polyhedron given by the convex hull of the Hilbert basis elements.     

On the Hilbert function of Artinian local complete intersections of codimension three

In singularity theory or algebraic geometry, it is natural to investigate possible Hilbert functions for special algebras A such as local complete intersections or more generally Gorenstein algebras. The sequences that occur as the Hilbert functions of standard graded complete intersections are well understood classically thanks to Macaulay and Stanley. Very little is known in the local case except in codimension two. In this paper we characterise the Hilbert functions of quadratic Artinian complete intersections of codimension three. Interestingly we prove that a Hilbert function is admissible for such a Gorenstein ring if and only if is admissible for such a complete intersection. We provide an effective construction of a local complete intersection for a given Hilbert function. We prove that the symmetric decomposition of such a complete intersection ideal is determined by its Hilbert function.     

Mixed multiplicities,Hilbert polynomials and homaloidal surfaces

We investigate the relationship among several numerical invariants associated to a (free) projective hypersurface V : the sequence of mixed multiplicities of its Jacobian ideal, the Hilbert polynomial of its Milnor algebra, and the sequence of exponents when V is free. As a byproduct, we obtain explicit equations for some of the homaloidal surfaces in the projective 3‐dimensional space constructed by C. Ciliberto, F. Russo and A. Simis.     

Galois representations for holomorphic Siegel modular forms

We prove local–global compatibility (up to a quadratic twist) of Galois representations associated to holomorphic Hilbert–Siegel modular forms in many cases (induced from Borel or Klingen parabolic), and as a corollary we obtain a conjecture of Skinner and Urban. For Siegel modular forms, when the local representation is an irreducible principal series we get local–global compatibility without a twist. We achieve this by proving a version of rigidity (strong multiplicity one) for GSp(4) using, on the one hand the doubling method to compute the standard L-function, and on the other hand the explicit classification of the irreducible local representations of GSp(4) over p-adic fields; then we use the existence of a globally generic Hilbert–Siegel modular form weakly equivalent to the original and we refer to Sorensen (Mathematica 15:623–670, 2010) for local–global compatibility in that case.