Dr Jim Shank
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 pgroups. 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 pgroups and finite subgroups of field. Journal of Pure and Applied Algebra [Online] 223:20152035. 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 pgroup. 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] 368:56555673. 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 pgroups of rank four. Involve: A Journal of Mathematics [Online] 9:551581. 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 pgroup 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 pgroups. Transformation Groups [Online] 18:122. Available at: http://dx.doi.org/10.1007/s000310139207z.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] 215:24812485. 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 psubgroups 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] 333:241257. 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] 225:10691094. 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 2dimensional 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(p1), 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_pmodules. Finally, noting that our representation of C_p on V_2 is as the pSylow 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] 207:149155. 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] 205:210225. 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 threedimensional 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] 357:36053621. Available at: http://dx.doi.org/10.1090/S0002994704035913.Let G be a finite group acting linearly on a finitedimensional 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
psubgroup for G. De. ne the cohomological connectivity of the
symmetric algebra S( V *) to be the smallest positive integer m such
that Hm( G, S( V *)) not equal 0. We show that min {dim(K)(VP) + 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 pnilpotent and P is cyclic, then,
for any modular representation, the depth of S( V *) G is min
{dim(K)(VP) + 2, dim(K)(V)}. 
Shank, R., Fleischmann, P. and Kemper, G. (2004). On the depth of cohomology modules. Quarterly Journal of Mathematics [Online] 55:167184. Available at: http://dx.doi.org/10.1093/qmath/hag046.We study the cohomology modules Hi(G,R) of a pgroup G acting on a
ring R of characteristic p, for i>0. In particular, we are interested
in the CohenMacaulay property and the depth of Hi(G,R) regarded as an
RGmodule. We first determine the support of Hi(G,R), which turns out
to be independent of i. Then we study the CohenMacaulay property for
H1(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 Hi(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 TrG R >RG has nonzero image. 
Fleischmann, P. and Shank, R. (2003). The relative trace ideal and the depth of modular rings of invariants. Archiv der Mathematik [Online] 80:347353. Available at: https://doi.org/10.1007/s0001300307940.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 pSylow 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 pgroups. Journal of Symbolic Computation [Online] 34:307327. Available at: http://dx.doi.org/10.1006/jsco.2002.0558.Let V be a finite dimensional representation of a pgroup, 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 U3 is the pSylow subgroup of GL(3)(Fp) 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 CohenMacaulay for p > 2. We also give a minimal generating set for k[mV(2)](Z/p) were V2 is the twodimensional 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] 34:438450. Available at: http://dx.doi.org/10.1112/S0024609302001054.Let W be a finitedimensional Z/pmodule 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 V3](Z/P),
where Vn is the indecomposable Z/pmodule of dimension n. 
Campbell, E., Hughes, I., Kemper, G., Shank, R. and Wehlau, D. (2000). Depth of modular invariant rings. Transformation Groups [Online] 5:2134. Available at: http://dx.doi.org/10.1007/BF01237176.It is wellknown that the ring of invariants associated to a neamodular representation of a finite group is CohenMacaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be CohenMacaulay 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). NonCohenMacaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants. Canadian Mathematical BulletinBulletin Canadien De Mathematiques 42:155161.This paper contains two essentially independent results in the invariant theory of finite groups. First
we prove that, for any faithful representation of a nontrivial pgroup over a field of characteristic p, the ring
of vector invariants ofmcopies of that representation is not CohenMacaulay 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 nontrivial 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. 105138. 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. 169196. Available at: http://dx.doi.org/10.1007/9780817648756_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. 241250. Available at: http://dx.doi.org/10.1090/crmp/035/17.
Thesis

Parsons, C. (2018). Modular Representations and Invariants of Elementary Abelian PGroups.The purpose of this thesis is to develop tools to more easily classify the modular representations of elementary abelian pgroups 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 pgroup 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 pgroups 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 psubgroups 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 longexploited 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 fourdimensional modular elementary abelian pgroups with rank 2, providing their invariant rings if CohenMacaulay 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 kvector 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 tworow groups, twocolumn groups, hook groups or one of two types of exceptional group.
The well known theorem of ChevalleyShephardTodd 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 pgroup and the invariant ring k[V ] G is CohenMacaulay then G is generated by bireflections. We use our classification to investigate which groups consisting of bireflections have CohenMacaulay 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 nonabelian hook groups, which have complete intersection invariant rings.
It is well known that CohenMacaulay invariant rings of pgroups 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 pgroup which has CohenMacaulay 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.