Some papers

Here are some papers you can download. Some are published research, some are unrefereed research, some are expository papers. They are loosely categorized as

Representation theory
1. Invariant distributions on n-fold metaplectic covers of p-adic GL(r,F) dvi , pdf , ps .
  Slightly shorter version appeared in J. Fourier Analysis and Applications, vol 7, 2001, 343-358. Describes the unitary and tempered dual of the n-fold metaplectic covers of SL(2,F), where F is a p-adic field with p not dividing 2n. Shows that any invariant distribution on such covers of SL(2,F) or of GL(on on such covers of SL(2,F) or of GL(he tempered dual.
2. On finite dimensional representations of non-connected reductive groups, J. of Lie Theory. 10(2000)269-284. At JLT web page
  Extends the classification of irreducible finite dimensional representations of almost simple algebraic groups over an algebraically closed field of characteristic zero to certain non-connected groups G where the component group is cyclic. Also extends some of Steinberg's results on the adjoint quotient G ---> T/W to these non-connected groups. These results are used to describe the geometry of twisted conjugacy classes of G^o, with respect to an automorphism of the connected group G^o. As an application: shows there is a "functorial" correspondence between virtual (finite dimensional) characters of twisted-invariant representations of G and virtual characters of an endoscopic group H of G.
3. On twisted orbital integral identities for PGL(3) over a p-adic field. dvi, ps , pdf .
  This appeared in the Canadian Journal of Math, vol 42, 1990, pp. 1098-1130. The object of this paper is to prove certain p-adic orbital integral identities needed in order to accomplish the symmetric square transfer via the twisted Arthur trace formula. Latexed 3-2003, correcting minor typos, with some minor changes to grammar and the construction of the figures.
4. Simple local trace formulas for unramified p-adic groups. dvi, ps , pdf .
  A corrected and slightly expanded version of a paper which appeared in Pac J Math, 1992, vol 154. The published version is full of typos. An errata list in dvi form of the published paper is at: ltr_err.dvi .
  This paper generalizes Waldspurger's method of deriving a local trace formula for spherical functions on GL(n) (J.-L. Waldspurger, Int\'egrales orbitales sph\'eriques pour GL(N) sur un corps p-adique, Asterique, vol 171-172, 1989) to any unramified group. A little later, Arthur published a vast generalization, replacing G by any connected reductive group and allowing smooth functions rather than only sphunctions rather than only sphi>A local trace formula, Pub Math IHES, 1991). Philosophically, Arthur's method is a generalization of Waldspurger's but technically speaking the proofs in this paper do not seem to extend to yield Arthur's local trace formula.
5. A hitch-hiker's guide to invariant distributions on SL(2) over a p-adic field dvi , pdf , ps .
  An expository paper on the invariant distributions on the locally compact group SL(2,F), where F is a p-adic field. Written 1994.
6. A hitch-hiker's guide to genuine invariant distributions on metaplectic covers of SL(2) over a p-adic field dvi, pdf, ps.
  An expository paper on the invariant distributions on the n-fold metaplectic cover of SL(2,F), where F is a p-adic field. Written 1994
7. On the unitary dual of SL(2,F)^-, F p-adic dvi , pdf , ps , 1999 preprint.
  Summarizes what is known about the unitary dual of the metaplectic covers of p-adic SL(2).
8. Notes on trace formulas for finite groups. dvi , pdf , ps .
  This is a paper which derives analogs of the global (Selberg) trace formula and the local (Arthur) trace formula for finite groups.
9. "A correspondence for the generalized Hecke algebra of the metaplectic cover of SL(2,F), F p-adic," New York J of Math. 4 (1998) 223-235. On the NYJM web page
10. (with David Jabon) On stable characters of tamely ramified supercuspidal representations of small unitary groups dvi , pdf , ps ,
  This is a much longer version of a paper in Far East J. of Math. Sci., 1 (1999)443-454. The motivation for this note lies in the following natural question for a the discrete series representation of a p-adic reductive group: is there a purely local algorithm which will produce its L-packet? As a means of producing an L-packet, the behaviour of the "stabilized character" of a discrete series representation pi of G is examined in the special case of the (unramified) unitary groups G=U(1,1)(F), U(2)(F), and U(2,1)(F), F a p-adic field, p>3.
11. (with Will Traves) Representations of finite groups on Riemann-Roch spaces dvi , pdf , ps Studies the action of a finite group on the Riemann-Roch space of certain divisors on a curve. If G is a finite subgroup of the automorphism group of a projective curve X over a field of "good" characteristic and D is a divisor on X stable by G then we show the natural representation of G on Riemann-Roch space L(D)=L_X(D) is a direct sum of irreducible representations of dimension <= d, where d is the size of the smallest G-orbit acting on X. We give an example to show that this is sharp (i.e., that dimension d subrepresentations do occur). Connections with coding theory, especially permutation decoding of AG codes, are given in the last section. Many examples are included.
12. (with Amy Ksir) Representations of finite groups on Riemann-Roch spaces, II dvi , pdf , ps Continues the studies in the first part. If $G$ is a finite subgroup of the automorphism group of a projective curve $X$ and $D$ is a divisor on $X$ stabilized by $G$, then under the assumption that $D$ is nonspecial, we compute a simplified formula for the trace of the natural representation of $G$ on Riemann-Roch space $L(D)$.
13. Representations of finite groups on the Riemann-Roch space of a toric variety, dvi , pdf , ps
  Studies the action of a finite group on the Riemann-Roch space of certain divisors on a toric variety X. If G is a finite subgroup of the automorphism group of X and D is a divisor on X stable by G then we show the natural representation of G on Riemann-Roch space L(D)=L_X(D) is a direct sum of permutation representations determined by the orbit of G on the polytope associated to X.
14. Modular representations on Riemann-Roch spaces: computational aspects , dvi , pdf , ps
  Notes for a talk given at Clemson University, October 23, 2003. Surveys some recent work on computing modular representations on Riemann-Roch spaces. No proofs. Last revised May 2005.
15. Modular representations on Riemann-Roch spaces: computational aspects , dvi , pdf , ps
16. (with Daniel Bump) L-functions for G₂ , dvi , pdf , ps
  This note written in 1987 presents an application of the Casselman--Shalika formula to local L-functions for the exceptional group G₂. More recently, David Ginzburg discovered a global integral which "unfolds" to our local integral. See his paper, "On the standard L-function for G₂," Duke J. Math. 69(1993)315-333.
17. (with Amy Ksir) Modular representations on some Riemann-Roch spaces of modular curves X(N) , pdf , ps
  This appeared in Computational Aspects of Algebraic Curves, Editor: T. Shaska, Lecture Notes in Computing, WorldScientific, 2005. However, we found one typo (the case N=13 mod 24 on page 21) which is corrected in this version.
  We compute the $PSL(2,N)$-module structure of the Riemann-Roch space L(D), where D is an invariant non-special divisor on the modular curve X(N), with N > 5 prime. This depends on a computation of the ramification module, which we give explicitly. These results hold for characteristic p if X(N) has good reduction mod p and p does not divide the order of PSL(2,N). We give as examples the cases N=7, 11, which were also computed using GAP. Applications to AG codes associated to this curve are considered.

Algebraic Curves
1. with Bob Guralnik and Daniel Goldstein, ``A Question about Pic(X) as a G-module,'' is available at math arxivs.
  Let G be a finite group acting faithfully on an irreducible non-singular projective curve defined over an algebraically closed field F. This paper addresses the following question: Does every G-invariant divisor class contain a G-invariant divisor? The answer depends only on G and not on the curve. We answer the same question for degree 0 divisor (classes).
  (A slightly different version will appear in Computational Aspects of Algebraic Curves, Editor: T. Shaska, Lecture Notes in Computing, WorldScientific, 2005.)
2. "Riemann-Roch space representations from bad hyperelliptic curves: questions and computations".
  Rough notes for a talk given at University of Idaho, May 27, 2005. Some problems on computing modular representations on Riemann-Roch spaces. No proofs.
  Beamer slides for the talk. latex , pdf .
Group theory
1. Arithmetic of characters of generalized symmetric groups . dvi , pdf , ps .
  A version of this appeared in Archiv der Mathematik, vol 81 (2003)113--120. (Copyright Springer-Verlag, 2003.)
  The result here answers the following questions in the affirmative: Can the Galois action on all abelian (Galois) fields K/Q be realized explicitly via an action on characters of some finite group? Are all subfields of a cyclotomic field of the form $Q (r)$, for some irreducible character r of a finite group G? In particular, we explicitly determine the Galois action on all irreducible characters of the generalized symmetric groups. We also explicitly determine the "smallest" extension of Q required to realize (using matrices) a given irreducible representation of a generalized symmetric group.
  Here is a more elementary longer version , with more examples and details.
2. Notes on splitting 2-fold covers of some "odd" unitary groups over a p-adic field. dvi , pdf , ps .
  This paper collects some remarks relating to works of Moen, Kudla, Kazhdan, and Li for 2-fold metaplectic covers quasi-split unitary groups in an odd number of variabry groups in an odd number of variabess a question raised in a letter from Gelbart to Moen. It is known that the metaplectic C¹-cover of these groups splits. Written 1995.
3. Which S_m have a complement in S_n?. dvi , ps , pdf (12 pages).
  This is an expository paper which classifies which symmetric groups S_m have a complement in S_n, for m less than n.
4. (with Pablo Legarraga) Smoothness Properties of Varieties of Borels and Schubert Varieties, (submitted). dvi , ps , pdf
  A similar version (using different methods) under the title "On the variety of Borels in relative position w" is available at math arxivs, This paper, with some modifications, appeared in Revista UMA, vol. 45, 2004, pp 69-74.
5. Introduction to group cohomology and homology . ps , pdf
  These are expanded lecture notes of a series of expository talks surveying basic aspects of group cohomology and homology.

Papers with students
1. (with Michael Fourte) Odd king tours on a chessboard dvi , pdf , ps .
  This paper shows, in particular, that the ordinary 8x8 chess board has no complete odd king tour, partially answering a question raised by C. Bailey and M. Kidwell in "A king's tour of the chessboard", Math. Mag.\underline{58}(1985)285-286. To appear in J. Rec. Math, volume 31, 2003, pp 173-177 Written 1997.
2. Ann Luers' (now Ann Casey) honors thesis The dodecahedral faces of M₁₂
3. (with Ann Luers) Mathematical blackjack using Steiner systems ,
4. Justin Montague's honors thesis "Searching with lies" ,
5. Aron Foster's honors thesis "A polynomial-time probabilistic algorithm for the minimum distance of an arbitrary linear error-correcting code"
6. Wayne Irons' thesis, "A polynomial-time probabilistic algorithm for the minimum distance of an arbitrary linear non-binary error-correcting code".
7. Clifton Lennon's thesis, "List-Decoding of Generalized Reed-Solomon Codes Using Sudan's Algorithm".
8. Jason McGowan's thesis, "Implementing Generalized Reed-Solomon Codes and a Cyclic Code Decoder in GUAVA".
9. Stephen McMath's Trident thesis, "Parallel Integer Factorization Using Quadratic Forms"
10. (with S. McMath, R. Crabbe) "Continued fractions and Parallel SQUFOF", pdf.
  This paper focuses on "square forms factorization" (SQUFOF), a method developed by Daniel Shanks in the 1970's. It is still the fastest known algorithm for factoring integers in the $20$- to $30$-digit range. This paper contains results:
  Though the results have not appeared in the literature in this form, these are known to the experts.
11. (with Jim McShea) m-ary Gray codes using MAPLE.
12. (with Jim McShea) Homology group of the square 1 puzzle - see section 13.5 in the book Adventures in group theory , and the errata.

Computational mathematics
1. (with Roland Martin) A MAPLE package for the decomposition of certain tensor products of Lie algebra representations using crystal graphs dvi , pdf
  A paper on a MAPLEV3,V4,V5 shareware package on decomposing tensor products representations of simple Lie algebras using crystal bases. See crystal for more details. See also the related paper dvi , pdf , also with Roland Martin.
2. Simple exercises in MAGMA dvi , ps , html .
  A paper on how to use MAGMA written for the first time user.
3. Representations of A₅ using MAGMA dvi , pdf , ps
  A paper on how to use MAGMA to analyze representations of finite groups, concentrating on A₅=PSL(2,5)=SL(2,4). Written for the first-time MAGMA user.
4. Representations of A₅ using GAP ps pdf .
  A paper on how to use GAP to analyze representations of finite groups, concentrating on A₅=PSL(2,5)=SL(2,4). Written for the first-time GAP user. Updated 2003.
5. (with Helena Verrill) Notes on toric varieties
  dvi , ps (360K), pdf (420K).
  Survey of most basic results on toric varieties. The emphasis is on the explicit desingularization of affine toric varieties, Riemann-Roch spaces of projective toric varieties, and (error-correcting) toric codes.
  MAGMA 2.8 code used in the paper: toric.mag, GAP 4.3 (requires the coding theory package GUAVA) code referred to in the paper: toric.g.
  A version of this paper will appear in Journal of Symbolic Computation (2007).
6. OSCAS - Maxima
  pdf .
  Communications in Computer Algebra (SIGSAM newsletter) article on Maxima.

Coding theory
1. (with Salahoddin Shokranian) Remarks on codes from modular curves: MAPLE applications. dvi , pdf , ps (12 pages).
  See also the MAGMA version at the math arXivs.
  This paper is an exposition of some aspects of geometric coding theory and Goppa codes on modular curves, which appeared in the book Coding Theory and Cryptography: From the Geheimschreiber and Enigma to Quantum Theory (ed. D. Joyner), Springer-Verlag. This paper is � Springer-Verlag (posted by permission).
2. Toric codes over finite fields. A version may be found at math arxivs .
  In this note, a class of error-correcting codes is associated to a toric variety defined over a finite field GF(q), analogous to the class of AG codes associated to a curve. For small q, many of these codes have parameters beating the Gilbert-Varshamov bound. In fact, using toric codes, we construct a [n,k,d]=[49,11,28] code over GF(8), which was (at the time of discovery) better than any other known code listed in Brouwer's tables for that n, k and q. We give upper and lower bounds on the minimum distance. We conclude with a discussion of some decoding methods. Many examples are given throughout.
  This paper is � Springer-Verlag (posted by permission). To appear in a 2004 issue of AAECC a Springer-Verlag journal, (DOI) 10.1007/s00200-004-0152-x AAECC (2004).
3. (with Amy Ksir) Automorphism groups of some AG codes pdf , ps . Studies the action of a finite group on the Riemann-Roch space of certain divisors on a specific hyperelliptic curve X defined over a finite field with "large" automorphism group G. If D is a divisor on X left stable by G then the natural representation of G on the Riemann-Roch space L(D). If E=P1+...+P_n is another divisor on X left stable by G then this representation on L(D) induces a permutation representation on the associated AG code C=C(D,E). This note discusses the permutation decoding of these AG codes. The main "results" are conjectures regarding the complexity of the permutation decoding of these hyperelliptic codes.
4. Introduction to codes from a representation-theoretic perspective hypertext dvi , pdf . This expository paper deals with some selected topics belonging to the intersection of the theory of error-correcting codes and the representation theory of finite groups. We shall see that codes which exhibit unusual symmetry often times turn out to be very interesting objects of study.
  Notes of lectures given to undergraduate math majors at Harvey Mudd College, May 2005.
5. Conjectural permutation decoding of some AG codes pdf . This appeared Communications in Computer Algbera Volume 39, March 2005, pages 166-172.
  We study the action of a finite group on the Riemann-Roch space of certain divisors on a specific hyperelliptic curve X defined over a finite field with "large" automorphism group G. This note discusses the permutation decoding on algebraic geometric codes C=C(D,E), where D and E are G-equivariant divisors on X. The main "results" are conjectures regarding the complexity of the permutation decoding of these hyperelliptic codes. The open source GAP error-correcting codes package GUAVA is used to compute examples.
6. (with Greg Coy) Goppa's conjecture and counting points on hyperelliptic curves pdf .
  A long standing problem has been to develop ``good" binary linear codes to be used for error-correction. We show in this paper that the Goppa conjecture regarding ``good'' binary codes is incompatible with a conjecture on the number of points of hyperelliptic curves over finite fields of odd prime order.
7. (with Amy Ksir and Will Traves) Automorphism groups of GRS codes pdf .
  We look at AG codes associated to P¹, re-examining the problem of determining their automorphism groups (originally investigated by Dur in 1987 using combinatorial techniques) using recent methods from algebraic geometry. We classify those finite groups that can arise as the automorphism group of an AG code and give an explicit description of how these groups appear. We give examples of generalized Reed-Solomon codes with large automorphism groups G, such as G=PSL(2,q), and explicitly describe their G-module structure.

Other
1. Principal Ideals and Associate Rings (with G. Benkart, A. Galione, M. Kidwell, M. Meyerson, D. Spellman, B. Wardlaw) dvi , pdf , ps .
  Appeared in Journal of Algebra, number theory, and applications 2(2002)181-193. A commutative ring A with 1 is associate provided whenever two elements a and b generate the same principal ideal there is a unit u such that ua=b. The main results proved here are:
  One novel feature here is that we prove these results using model theory. The main authors are Gaglione and Spellman.
2. A note on gaps between zeroes of L-functions dvi , ps, pfd .
  A paper on the vertical distribution of zeroes of certain L-functions. Written 1984-5, slightly revised 1995.
3. On the Kuznetsov-Bruggeman formula for a Hilbert modular surface having one cusp. dvi , ps , pdf .
  This is a revised and corrected 1998 version of a paper which appeared in Math Zeit., 1990, vol 203. 54 pages.
4. The Mathemataka.
  Math humor (a satirical mathematical analog of the famous Loginataka from computer science) dvi , pdf , ps .

wdj@usna.edu

Last updated 3-30-2007.

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