About

Jim completed his PhD in homotopy theory at the University of Toronto in 1989 under supervision of Professor Paul Selick. He has been an academic at the University of Kent since 1999. Jim's current research interests involve the invariant theory of finite groups and related aspects of commutative algebra and representation theory. 

Research interests

Jim's current research involves the invariant theory of finite groups and related aspects of commutative algebra and representation theory. He is particularly interested in the rings of invariants of modular representations of p-groups. Research collaborators at the University of Kent include Peter Fleischmann and Chris Woodcock. Overseas collaborators include Eddy Campbell and Jianjun Chuai at the University of New Brunswick, David Wehlau (Queen’s University/RMC, Kingston), Yin Chen (Northeast Normal University, Changchun) and Műfit Sezer (Bilkent University, Ankara).

Publications

Article

  • Campbell, H., Chuai, J., Shank, R. and Wehlau, D. (2018). Representations of elementary abelian p-groups and finite subgroups of field. Journal of Pure and Applied Algebra [Online] 223:2015-2035. Available at: https://dx.doi.org/10.1016/j.jpaa.2018.08.013.
    Suppose F is a field of prime characteristic p and E is a finite subgroup of the additive group (F,+). Then E is an elementary abelian p-group. We consider two such subgroups, say E and E', to be equivalent if there is an ? ? F× := F \ {0} such that E = ?E'. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants.
  • Sezer, M. and Shank, R. (2016). Rings of invariants for modular representations of the Klein four group. Transactions of the American Mathematical Society [Online] 368:5655-5673. Available at: http://dx.doi.org/10.1090/tran/6516.
    We study the rings of invariants for the indecomposable modular representations of the Klein four group. For each such representation we compute the Noether number and give minimal generating sets for the Hilbert ideal and the field of fractions. We observe that, with the exception of the regular representation, the Hilbert ideal for each of these representations is a complete intersection.
  • Shank, R. and Pierron, T. (2016). Rings of invariants for the three dimensional modular representations of elementary abelian p-groups of rank four. Involve: A Journal of Mathematics [Online] 9:551-581. Available at: http://dx.doi.org/10.2140/involve.2016.9.551.
    We show that the rings of invariants for the three
    dimensional modular representations of an elementary abelian p-group of rank four are
    complete intersections with embedding dimension at most five. Our results confirm
    the conjectures of Campbell, Shank and Wehlau
    for these representations.
  • Campbell, E., Shank, R. and Wehlau, D. (2013). Rings of invariants for modular representations of elementary abelian p-groups. Transformation Groups [Online] 18:1-22. Available at: http://dx.doi.org/10.1007/s00031-013-9207-z.
    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.
  • Hobson, A. and Shank, R. (2011). The invariants of the second symmetric power representation of SL_2(F_q). Journal of Pure and Applied Algebra [Online] 215:2481-2485. Available at: http://dx.doi.org/10.1016/j.jpaa.2011.02.006.
    For a prime p>2 and q=p^n, we compute a finite generating set for the SL_2(F_q)-invariants of the second symmetric power representation, showing the invariants are a hypersurface and the field of fractions is a purely transcendental extension of the coefficient field. As an intermediate result, we show the invariants of the Sylow p-subgroups are also hypersurfaces.
  • Shank, R. and Hobson, A. (2011). The invariants of the third symmetric power representation of SL_2(F_p). Journal of Algebra [Online] 333:241-257. Available at: http://dx.doi.org/10.1016/j.jalgebra.2011.02.023.
    For a prime p>3, we compute a finite generating set for the
    SL_2(F_p)-invariants of the third symmetric power representation. The proof relies on the construction of an infinite SAGBI basis and uses the Hilbert series calculation of Hughes and Kemper.
  • Campbell, E., Shank, R. and Wehlau, D. (2010). Vector invariants for the two dimensional modular representation of a cyclic group of prime order. Advances in Mathematics [Online] 225:1069-1094. Available at: http://dx.doi.org/10.1016/j.aim.2010.03.018.
    In this paper, we study the vector invariants, F[mV_2]^(C_p), of the 2-dimensional indecomposable representation V_2 of the cylic group, C_p, of order p over a field F of characteristic p. This ring of invariants was first studied by David Richman who showed that this ring required a generator of degree m(p-1), thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p=2. This conjecture was proved by Campbell and Hughes. Later, Shank and Wehlau determined which elements in Richman's generating set were redundant thereby producing a minimal generating set.

    We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants F[m V_2]^(C_p). In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis. Further, our techniques also serve to give an explicit decomposition of F[m V_2] into a direct sum of indecomposable C_p-modules. Finally, noting that our representation of C_p on V_2 is as the p-Sylow subgroup of SL_2(F_p), we are able to determine a generating set for the ring of invariants of F[m V_2]^(SL_2(F_p)).
  • Shank, R., Fleischmann, P., Sezer, M. and Woodcock, C. (2006). The Noether numbers for cyclic groups of prime order. Advances in Mathematics [Online] 207:149-155. Available at: http://dx.doi.org/10.1016/j.aim.2005.11.009.
    The Noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants. We compute the Noether number for an arbitrary representation of a cyclic group of prime order, and as a consequence prove the "2p?3 conjecture."
  • Sezer, M. and Shank, R. (2006). On the coinvariants of modular representations of cyclic groups of prime order. Journal of Pure and Applied Algebra [Online] 205:210-225. Available at: http://dx.doi.org/doi:10.1016/j.jpaa.2005.07.003.
    We consider the ring of coinvariants for modular representations of cyclic groups of prime order. For all cases for which explicit generators for the ring of invariants are known, we give a reduced Gröbner basis for the Hilbert ideal and the corresponding monomial basis for the coinvariants. We also describe the decomposition of the coinvariants as a module over the group ring. For one family of representations, we are able to describe the coinvariants despite the fact that an explicit generating set for the invariants is not known. In all cases our results confirm the conjecture of Harm Derksen and Gregor Kemper on degree bounds for generators of the Hilbert ideal. As an incidental result, we identify the coefficients of the monomials appearing in the orbit product of a terminal variable for the three-dimensional indecomposable representation.
  • Fleischmann, P., Kemper, G. and Shank, R. (2005). Depth and cohomological connectivity in modular invariant theory. Transactions of the American Mathematical Society [Online] 357:3605-3621. Available at: http://dx.doi.org/10.1090/S0002-9947-04-03591-3.
    Let G be a finite group acting linearly on a finite-dimensional vector
    space V over a field K of characteristic p. Assume that p divides the
    order of G so that V is a modular representation and let P be a Sylow
    p-subgroup for G. De. ne the cohomological connectivity of the
    symmetric algebra S( V *) to be the smallest positive integer m such
    that H-m( G, S( V *)) not equal 0. We show that min {dim(K)(V-P) + m+
    1, dim(K)( V)} is a lower bound for the depth of S( V *) G. We
    characterize those representations for which the lower bound is sharp
    and give several examples of representations satisfying the criterion.
    In particular, we show that if G is p-nilpotent and P is cyclic, then,
    for any modular representation, the depth of S( V *) G is min
    {dim(K)(V-P) + 2, dim(K)(V)}.
  • Shank, R., Fleischmann, P. and Kemper, G. (2004). On the depth of cohomology modules. Quarterly Journal of Mathematics [Online] 55:167-184. Available at: http://dx.doi.org/10.1093/qmath/hag046.
    We study the cohomology modules H-i(G,R) of a p-group G acting on a
    ring R of characteristic p, for i>0. In particular, we are interested
    in the Cohen-Macaulay property and the depth of H-i(G,R) regarded as an
    R-G-module. We first determine the support of H-i(G,R), which turns out
    to be independent of i. Then we study the Cohen-Macaulay property for
    H-1(G,R). Further results are restricted to the special case that G is
    cyclic and R is the symmetric algebra of a vector space on which G
    acts. We determine the depth of H-i(G,R) for i odd and obtain results
    in certain cases for i even. Along the way, we determine the degrees in
    which the transfer map Tr-G R -->R-G has non-zero image.
  • Fleischmann, P. and Shank, R. (2003). The relative trace ideal and the depth of modular rings of invariants. Archiv der Mathematik [Online] 80:347-353. Available at: https://doi.org/10.1007/s00013-003-0794-0.
    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.
  • Shank, R. and Wehlau, D. (2002). Computing modular invariants of p-groups. Journal of Symbolic Computation [Online] 34:307-327. Available at: http://dx.doi.org/10.1006/jsco.2002.0558.
    Let V be a finite dimensional representation of a p-group, G, over a field, k, of characteristic p. We show that there exists a choice of basis and monomial order for which the ring of invariants, k[V](G), has a finite SAGBI basis. We describe two algorithms for constructing a generating set fork[V]G. We use these methods to analyse k[2V(3)](U3) where U-3 is the p-Sylow subgroup of GL(3)(F-p) and 2V(3) is the sum of two copies of the canonical representation. We give a generating set for k[2V(3)](U3) for p = 3 and prove that the invariants fail to be Cohen-Macaulay for p > 2. We also give a minimal generating set for k[mV(2)](Z/p) were V-2 is the two-dimensional indecomposable representation of the cyclic group Z/p. (C) 2002 Elsevier Science Ltd.. All rights reserved.
  • Shank, R. and Wehlau, D. (2002). Noether numbers for subrepresentations of cyclic groups of prime order. Bulletin of the London Mathematical Society [Online] 34:438-450. Available at: http://dx.doi.org/10.1112/S0024609302001054.
    Let W be a finite-dimensional Z/p-module over a field, k, of
    characteristic p. The maximum degree of an indecomposable element of
    the algebra of invariants, k[W](Z/P), is called the Noether number of
    the representation, and is denoted by beta(W). A lower bound for
    beta(W) is derived, and it is shown that if U is a Z/p submodule of W,
    then beta(U) less than or equal to beta(W). A set of generators, in
    fact a SAGBI basis, is constructed for k[V2 circle plus V-3](Z/P),
    where V-n is the indecomposable Z/p-module of dimension n.
  • Campbell, E., Hughes, I., Kemper, G., Shank, R. and Wehlau, D. (2000). Depth of modular invariant rings. Transformation Groups [Online] 5:21-34. Available at: http://dx.doi.org/10.1007/BF01237176.
    It is well-known that the ring of invariants associated to a nea-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 paper(1) 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. Tn particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [6].
  • Campbell, E., GeramitaI, A., Hughes, I., Wehlau, D. and Shank, R. (1999). Non-Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants. Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques 42:155-161.
    This paper contains two essentially independent results in the invariant theory of finite groups. First
    we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring
    of vector invariants ofmcopies of that representation is not Cohen-Macaulay form 3. In the second section
    of the paper we use Poincar´e series methods to produce upper bounds for the degrees of the generators for
    the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and
    a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is
    generated in degrees less than or equal to n(jGj ? 1). If the ring of invariants is a hypersurface, the upper
    bound can be improved to [G].

Book section

  • Fleischmann, P. and Shank, R. (2016). The Invariant Theory of Finite Groups. In: Bullett, S., Fearn, T. and Smith, F. eds. Algebra, Logic and Combinatorics. London, UK: World Scientific, pp. 105-138. Available at: http://www.worldscientific.com/worldscibooks/10.1142/q0009.
    Mathematicians seek to exploit all available symmetry and often encode symmetry using the language of group actions. In this chapter we consider finite groups acting by ring automorphisms on a polynomial ring. Our goal is to understand the subring of invariant polynomials.
  • Shank, R. and Wehlau, D. (2009). Decomposing symmetric powers of certain modular representations of cyclic groups. In: Campbell, E., Helminck, A. G., Kraft, H. and Wehlau, D. L. eds. Symmetry and Spaces: In Honor of Gerry Schwarz. Berlin: Birkhauser, pp. 169-196. Available at: http://dx.doi.org/10.1007/978-0-8176-4875-6_9.
    For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p^2. We then use the constructed invariants to describe the decomposition of the symmetric algebra as a module over the group ring, confirming the Periodicity Conjecture of Ian Hughes and Gregor Kemper for this case.
  • Shank, R. (2004). Classical covariants and modular invariants. In: Campbell, E. and Wehlau, D. L. eds. Invariant Theory in All Characteristics. American Mathematical Society, pp. 241-250. Available at: http://dx.doi.org/10.1090/crmp/035/17.

Thesis

  • Parsons, C. (2018). Modular Representations and Invariants of Elementary Abelian P-Groups.
    The purpose of this thesis is to develop tools to more easily classify the modular representations of elementary abelian p-groups and better understand their invariant rings. Since these groups almost always have wild representation type complete classification of the indecomposables is considered impossible and as such an alternative perspective is required.
    We reformulate the representation classification in the perspective of classifying maximal abelian subgroups of unipotent groups. Thence we express the problem as determining finitely many 'covering' homomorphisms of the form ? : (F^d, +) ? GL_n(F) whose images collectively contain the images of all such representations up to
    equivalence. To aid in this we attach a combinatorial equivalence invariant object to modular p-group representations thereby allowing us to segment the problem and more easily distinguish between inequivalent families.
    Using these tools we build upon the work of [11] and develop a full set of covering homomorphisms for all modular elementary abelian p-groups in GL_4(F), GL_5(F) and GL_6(F). In doing so we also provide covering
    homomorphisms for select families in arbitrary dimension with specific patterns in their combinatorial invariant. By way of example we use these to provide an explicit construction for the Sylow p-subgroups of the finite orthogonal groups.
    Thereafter our focus switches to invariant rings. Given a matrix group in
    the image of a homomorphism ? : (F^d, +) ? GL_n(F) we explore methods of recovering the W ? (F^d, +) used to generate the group purely through its action on specific elements in the symmetric algebra of the dual, properties of which are indicative of properties of the invariant ring. Using this we provide an alternative explicit construction for the invariant rings implicitly generated in [8]. Further we generalise a long-exploited technique for inductively defining invariants from those of maximal subgroups. After classifying the invariant rings and fields of several hitherto classified families we focus on the four-dimensional modular elementary abelian p-groups with rank 2, providing their invariant rings if Cohen-Macaulay and algorithmic methods to procure it otherwise.
  • Horan, K. (2017). On the Invariant Theory of Finite Unipotent Groups Generated by Bireflections.
    Let k be a field of characteristic p and let V be a k-vector space. In Chapter 2 of this thesis we classify all unipotent groups G ? GL(V ) consisting of bireflections for p not equal to 2: we show that unipotent groups consisting of bireflections are either two-row groups, two-column groups, hook groups or one of two types of exceptional group.
    The well known theorem of Chevalley-Shephard-Todd shows the importance of (pseudo-)reflection groups to invariant theory. Our interest in bireflection groups is motivated by the theorem of Kemper which tells us if G ? GL(V ) is a p-group and the invariant ring k[V ] G is Cohen-Macaulay then G is generated by bireflections. We use our classification to investigate which groups consisting of bireflections have Cohen-Macaulay or complete intersection invariant rings.
    In Chapter 3 we introduce techniques and notation which we use later to find invariant rings of groups by viewing them as subgroups of Nakajima groups. In Chapter 4 we show that for k = Fp there is a family of hook groups, including all non-abelian hook groups, which have complete intersection invariant rings.
    It is well known that Cohen-Macaulay invariant rings of p-groups in characteristic p are Gorenstein. There has been speculation by experts in the area, that they might in fact be complete intersections. In Chapter 5 we settle this negatively by giving an example of a p-group which has Cohen-Macaulay but non complete intersection invariant ring. To the best of our knowledge this is the first example of that kind.
    Finally in Chapter 6 we show that when k = F_p both types of exceptional group have complete intersection invariant rings.
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