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Paul E. Gunnells

Publications and source records attributed to Paul E. Gunnells.

At least 37 records · Page 2Linked to original sources
arXiv2014

Cells in Coxeter groups I

The purpose of this article is to shed new light on the combinatorial structure of Kazhdan-Lusztig cells in infinite Coxeter groups $W$. Our main focus is the set $\D$ of distinguished involutions in $W$, which was introduced by Lusztig in one of his first papers on cells in affine Weyl groups. We conjecture that the set $\D$ has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite Coxeter groups. Moreover, to each element of $\D$ we assign an explicitly defined set of equivalence relations on $W$ that altogether conjecturally determine the partition of $W$ into left (right) cells. We are able to prove these conjectures only in a special case, but even from these partial results we can deduce some interesting corollaries.

math.RT↗
arXiv2013

On the cohomology of linear groups over imaginary quadratic fields

Let Gamma be the group GL_N (OO_D), where OO_D is the ring of integers in the imaginary quadratic field with discriminant D<0. In this paper we investigate the cohomology of Gamma for N=3,4 and for a selection of discriminants: D >= -24 when N=3, and D=-3,-4 when N=4. In particular we compute the integral cohomology of Gamma up to p-power torsion for small primes p. Our main tool is the polyhedral reduction theory for Gamma developed by Ash and Koecher. Our results extend work of Staffeldt, who treated the case n=3, D=-4. In a sequel to this paper, we will apply some of these results to the computations with the K-groups K_4 (OO_{D}), when D=-3,-4.

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arXiv2013

Torus orbits on homogeneous varieties and Kac polynomials of quivers

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients.

math.RT↗
arXiv2013

Mod 2 homology for GL(4) and Galois representations

We extend the computations in [AGM4] to find the mod 2 homology in degree 1 of a congruence subgroup Gamma of SL(4,Z) with coefficients in the sharbly complex, along with the action of the Hecke algebra. This homology group is closely related to the cohomology of Gamma with F_2 coefficients in the top cuspidal degree. These computations require a modification of the algorithm to compute the action of the Hecke operators, whose previous versions required division by 2. We verify experimentally that every mod 2 Hecke eigenclass found appears to have an attached Galois representation, giving evidence for a conjecture in [AGM4]. Our method of computation was justified in [AGM5].

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arXiv2012

Defeating the Kalka--Teicher--Tsaban linear algebra attack on the Algebraic Eraser

The Algebraic Eraser (AE) is a public key protocol for sharing information over an insecure channel using commutative and noncommutative groups; a concrete realization is given by Colored Burau Key Agreement Protocol (CBKAP). In this paper, we describe how to choose data in CBKAP to thwart an attack by Kalka--Teicher--Tsaban.

cs.CR↗
arXiv2011

On Hilbert modular threefolds of discriminant 49

Let K be the totally real cubic field of discriminant 49, let O be its ring of integers, and let p be the prime over 7. Let Gamma (p)\subset Gamma = SL_2(O) be the principal congruence subgroup of level p. This paper investigates the geometry of the Hilbert modular threefold attached to Gamma (p) and some related varieties. In particular, we discover an octic in P^3 with 84 isolated singular points of type A_2.

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arXiv2011

Resolutions of the Steinberg module for GL(n)

We give several resolutions of the Steinberg representation St_n for the general linear group over a principal ideal domain, in particular over Z. We compare them, and use these results to prove that the computations in [AGM4] are definitive. In particular, in [AGM4] we use two complexes to compute certain cohomology groups of congruence subgroups of SL(4,Z). One complex is based on Voronoi's polyhedral decomposition of the symmetric space for SL(n,R), whereas the other is a larger complex that has an action of the Hecke operators. We prove that both complexes allow us to compute the relevant cohomology groups, and that the use of the Voronoi complex does not introduce any spurious Hecke eigenclasses.

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arXiv2011

On the cryptanalysis of the generalized simultaneous conjugacy search problem and the security of the Algebraic Eraser

The Algebraic Eraser (AE) is a cryptographic primitive that can be used to obscure information in certain algebraic cryptosystems. The Colored Burau Key Agreement Protocol (CBKAP), which is built on the AE, was introduced by I. Anshel, M. Anshel, D. Goldfeld, and S. Lemieux in 2006 as a protocol suitable for use on platforms with constrained computational resources, such as RFID and wireless sensors. In 2009 A. Myasnikov and A. Ushnakov proposed an attack on CBKAP that attempts to defeat the generalized simultaneous conjugacy search problem, which is the public-key computational problem underlying CBKAP. In this paper we investigate the effectiveness of this attack. Our findings are that success of the attack only comes from applying it to short keys, and that with appropriate keys the attack fails in 100% of cases and does not pose a threat against CBKAP. Moreover, the attack makes assumptions about CBKAP that do not hold in practical implementations, and thus does not represent a threat to the use of CBKAP in applications.

cs.CR↗
arXiv2010

Metaplectic Ice

Spherical Whittaker functions on the metaplectic n-fold cover of GL(r+1) over a nonarchimedean local field containing n distinct n-th roots of unity may be expressed as the partition functions of statistical mechanical systems that are variants of the six-vertex model. If n=1 then in view of the Casselman-Shalika formula this fact is related to Tokuyama's deformation of the Weyl character formula. It is shown that various properties of these Whittaker functions may be expressed in terms of the commutativity of row transfer matrices for the system. Potentially these properties (which are already proved by other methods, but very nontrivial) are amenable to proof by the Yang-Baxter equation.

math.RT↗
arXiv2010

Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels up to 31.

math.NT↗
arXiv2009

Littelmann patterns and Weyl group multiple Dirichlet series of type D

We formulate a conjecture for the local parts of Weyl group multiple Dirichlet series attached to root systems of type D. Our conjecture is analogous to the description of the local parts of type A series given by Brubaker, Bump, Friedberg, and Hoffstein in terms of Gelfand--Tsetlin patterns. Our conjecture is given in terms of patterns for irreducible representations of even orthogonal Lie algebras developed by Littelmann.

math.NT↗
arXiv2009

Perfect forms over totally real number fields

A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and later generalized by Koecher to arbitrary number fields. One knows that up to a natural "change of variables'' equivalence, there are only finitely many perfect forms, and given an initial perfect form one knows how to explicitly compute all perfect forms up to equivalence. In this paper we investigate perfect forms over totally real number fields. Our main result explains how to find an initial perfect form for any such field. We also compute the inequivalent binary perfect forms over real quadratic fields Q(\sqrt{d}) with d \leq 66.

math.NT↗
arXiv2009

Cohomology of Congruence Subgroups of SL_4(Z). III

In two previous papers [AGM1, AGM2] we computed cohomology groups H^5(Γ_0 (N); \C) for a range of levels N, where Γ_0 (N) is the congruence subgroup of SL_4 (\Z) consisting of all matrices with bottom row congruent to (0,0,0,*) mod N. In this note we update this earlier work by carrying it out for prime levels up to N = 211. This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20 million non-zero entries. We also make two conjectures concerning the contributions to H^5(Γ_0 (N); \C) for N prime coming from Eisenstein series and Siegel modular forms.

math.NT↗
arXiv2009

Constructing Weyl group multiple Dirichlet series

Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, that has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. A heuristic definition of such series was given in [2], and they have been investigated in certain special cases in [2-6, 11-14]. In this paper we generalize results in [13] to construct Weyl group multiple Dirichlet series by a uniform method, and show in all cases that they have the expected properties.

math.NT↗
arXiv2009

On the cohomology of congruence subgroups of SL_4 (\Z)

We survey our joint work with Avner Ash and Mark McConnell that computationally investigates the cohomology of conguence subgroups of SL_4 (\Z). This is based on a talk given at the 2009 RIMS conference "Automorphic representations, automorphic L-functions and arithmetic."

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