Portrait of Professor Peter Fleischmann

Professor Peter Fleischmann

Professor of Pure Mathematics
Director of Studies MMath



Research interests

  • Representation theory and structure theory of finite groups
  • Constructive invariant theory
  • Applied algebra
  • Discrete mathematics.



  • Horan, K. and Fleischmann, P. (2019). The finite unipotent groups consisting of bireflections. Journal of Group Theory [Online] 22:191-230. Available at: https://doi.org/10.1515/jgth-2018-0123.
    Let k be a field of characteristic p and V a finite dimensional k-vector space. An element g ? GL(V) is called a bireflection if it centralizes a subspace of codimension less than or equal to 2. It is known by a result of Kemper, that if for a finite p-group G ? GL(V) the ring of invariants Sym(V?) G is Cohen-Macaulay, G is generated by bireflections. Although the converse is false in general, it holds in special cases e.g. for particular families of groups consisting of bireflections. In this paper we give, for p > 2, a classification of all finite unipotent subgroups of GL(V) consisting of bireflections. Our description of the groups is given explicitly in terms useful for exploring the corresponding rings of invariants. This further analysis will be the topic of a forthcoming paper.
  • Fleischmann, P. and Woodcock, C. (2018). Free actions of p-groups on affine varieties in characteristic p. Mathematical Proceedings of the Cambridge Philosophical Society [Online] 165:109-135. Available at: https://doi.org/10.1017/S0305004117000317.
    Let K be an algebraically closed field and n ? K n affine n-space. It is known that a finite group can only act freely on n if K has characteristic p > 0 and is a p-group. In that case the group action is “non-linear” and the ring of regular functions K[ n ] must be a trace-surjective K ? -algebra.

    Now let k be an arbitrary field of characteristic p > 0 and let G be a finite p-group. In this paper we study the category of all finitely generated trace-surjective k ? G algebras. It has been shown in [13] that the objects in are precisely those finitely generated k ? G algebras A such that A G ? A is a Galois-extension in the sense of [7], with faithful action of G on A. Although is not an abelian category it has “s-projective objects”, which are analogues of projective modules, and it has (s-projective) categorical generators, which we will describe explicitly. We will show that s-projective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category also has “weakly initial objects”, which are closely related to the essential dimension of G over k. Our results yield a geometric structure theorem for free actions of finite p-groups on affine k-varieties. There are also close connections to open questions on retracts of polynomial rings, to embedding problems in standard modular Galois-theory of p-groups and, potentially, to a new constructive approach to homogeneous invariant theory
  • Fleischmann, P. and Woodcock, C. (2018). Stable transcendence for formal power series, generalized Artin-Schreier polynomials and a conjecture concerning p-groups. Bulletin of the London Mathematical Society [Online] 50:933-944. Available at: https://doi.org/10.1112/blms.12197.
    Let f(x) be a formal power series with coefficients in the field k
    and let n ? 1. We define the notion of n-transcendence of f(x) over k and, more
    generally, the stable transcendence function dk(f(x), n). It is shown that, if k
    has prime characteristic p, this function determines the minimal Krull dimension
    dk(G) of the universal modular Galois-algebras of an elementary Abelian
    p-group G, introduced in [2, 3, 4, 5]. Since the concept of n-transcendence is of
    independent interest in all characteristics, a number of fundamental theorems
    are proved where the generalized Artin-Schreier polynomials surprisingly play
    a central role. We make a plausible conjecture in the case when k = Fp, the
    truth of which would imply a conjectural result concerning dFp
    (G) previously investigated by the authors.
  • Ferreira, J. and Fleischmann, P. (2016). The invariant rings of the Sylow groups of GU(3,q2), GU(4,q2), Sp(4,q) and O+(4,q) in the natural characteristic. Journal of Symbolic Computation [Online] 79:356-371. Available at: http://dx.doi.org/10.1016/j.jsc.2016.02.013.
    Let G be a Sylow p -subgroup of the unitary groups GU(3,q2)GU(3,q2), GU(4,q2)GU(4,q2), the symplectic group Sp(4,q)Sp(4,q) and, for q odd, the orthogonal group O+(4,q)O+(4,q). In this paper we construct a presentation for the invariant ring of G acting on the natural module. In particular we prove that these rings are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant form defining the corresponding classical group. We also show that these generators form a SAGBI basis and the invariant ring for G is a complete intersection.
  • Fleischmann, P. and Ferreira, J. (2015). The invariant fields of the Sylow groups of classical groups in the natural characteristic. Communications in Algebra [Online] 44:977-1010. Available at: http://dx.doi.org/10.1080/00927872.2014.999922.
    Let X be any finite classical group defined over a finite field of characteristic p > 0. In
    this article, we determine the fields of rational invariants for the Sylow p-subgroups of
    X, acting on the natural module. In particular, we prove that these fields are generated
    by orbit products of variables and certain invariant polynomials which are images under
    Steenrod operations, applied to the respective invariant linear forms defining X.
  • Fleischmann, P. and Woodcock, C. (2015). Modular group actions on algebras and p-local Galois extensions for finite groups. Journal of Algebra [Online] 442:316-353. Available at: http://dx.doi.org/10.1016/j.jalgebra.2015.06.003.
    Let k be a field of positive characteristic p and let G be a finite group. In this paper we study the category TsGTsG of finitely generated commutative k-algebras A on which G acts by algebra automorphisms with surjective trace. If A=k[X]A=k[X], the ring of regular functions of a variety X, then trace-surjective group actions on A are characterized geometrically by the fact that all point stabilizers on X are p?p?-subgroups or, equivalently, that AP?AAP?A is a Galois extension for every Sylow p-group of G . We investigate categorical properties of TsGTsG, using a version of Frobenius-reciprocity for group actions on k -algebras, which is based on tensor induction for modules. We also describe projective generators in TsGTsG, extending and generalizing the investigations started in [7], [8] and [9] in the case of p-groups. As an application we show that for an abelian or p-elementary group G and k large enough, there is always a faithful (possibly nonlinear) action on a polynomial ring such that the ring of invariants is also a polynomial ring. This would be false for linear group actions by a result of Serre. If A is a normal domain and G?Autk(A)G?Autk(A) an arbitrary finite group, we show that AOp(G)AOp(G) is the integral closure of k[Soc(A)]k[Soc(A)], the subalgebra of A generated by the simple kG-submodules in A. For p-solvable groups this leads to a structure theorem on trace-surjective algebras, generalizing the corresponding result for p-groups in [8].
  • Fleischmann, P. and Woodcock, C. (2013). Universal Galois algebras and cohomology of p-groups. Journal of Pure and Applied Algebra [Online] 217:530-545. Available at: http://dx.doi.org/10.1016/j.jpaa.2012.06.023.
    Let G be a finite p-group and k a field of characteristic p>0. A universal Galois algebra of G is a weakly initial object in the category Ts of trace-surjective (commutative) k?G algebras. The objects in Ts are precisely the k?G-algebras that are Galois ring extensions over the ring of G-invariants. They are also characterized as k?G algebras which are free kG-modules. One example is Dk, the dehomogenized symmetric algebra of the regular representation, which is also an s-projective object in Ts (see the definition in the paper). In the previous work we proved that the polynomial ring Dk (of dimension ?G??1) contains a polynomial retract U?Ts of dimension , such that the invariant rings UG and are again polynomial rings. The G-action on U will in general be highly non-linear, but in special cases it can be chosen to be “almost linear”. In this paper we investigate such almost linear universal algebras, generalizing the construction of the algebras Dk and U. It is known that for k=Fp the minimal dimension of a polynomial universal algebra is n. Among other things we prove that such an algebra can be realized in an “almost linear” way, if and only if G is “crossed isomorphic” to an Fp-vector space. This is equivalent to the existence of a kG-module V such that there is 0?[?]?H1(G,V) with ??Z1(G,V) being a bijective cocycle.
  • Fleischmann, P. and Woodcock, C. (2013). Galois ring extensions and localized modular rings of invariants of p-groups. Transformation Groups [Online] 18:131-147. Available at: http://dx.doi.org/10.1007/s00031-012-9205-6.
    We apply recent results on Galois-ring extensions and trace surjective algebras to analyze dehomogenized modular invariant rings of finite p-groups, as well as related localizations. We describe criteria for the dehomogenized invariant ring to be polynomial or at least regular and we show that for regular affine algebras with possibly non-linear action by a p-group, the singular locus of the invariant ring is contained in the variety of the transfer ideal. If V is the regular module of an arbitrary finite p-group, or V is any faithful representation of a cyclic p-group, we show that there is a suitable invariant linear form, inverting which renders the ring of invariants into a "localized polynomial ring" with dehomogenization being a polynomial ring. This is in surprising contrast to the fact that for a faithful representation of a cyclic group of order larger than p, the ring of invariants itself cannot be a polynomial ring by a result of Serre. Our results here generalize observations made by Richman [R] and by Campbell and Chuai [CCH].
  • Fleischmann, P. (2013). On finite unipotent transvection groups and their invariants. Quarterly Journal of Mathematics [Online] 65:175-200. Available at: http://dx.doi.org/10.1093/qmath/has042.
  • Fleischmann, P. and Woodcock, C. (2011). Non-linear group actions with polynomial invariant rings and a structure theorem for modular Galois extensions. Proceedings of the London Mathematical Society [Online] 103:826-846. Available at: http://dx.doi.org/10.1112/plms/pdr016.
    Let G be a finite p-group and k be a field of characteristic p>0. We show that G has a non-linear faithful action on a polynomial ring U of dimension n=log p(|G|) such that the invariant ring UG is also polynomial. This contrasts with the case of linear and graded group actions with polynomial rings of invariants, where the classical theorem of Chevalley–Shephard–Todd and Serre requires G to be generated by pseudo-reflections. Our result is part of a general theory of ‘trace surjective G-algebras’, which, in the case of p-groups, coincide with the Galois ring extensions in the sense of Chase, Harrison and Rosenberg [‘Galois theory and Galois cohomology of commutative rings’, Mem. Amer. Math. Soc. 52 (1965) 15–33]. We consider the dehomogenized symmetric algebra Dk, a polynomial ring with non-linear G-action, containing U as a retract and we show that DGk is a polynomial ring. Thus, U turns out to be universal in the sense that every trace surjective G-algebra can be constructed from U by ‘forming quotients and extending invariants’. As a consequence we obtain a general structure theorem for Galois extensions with given p-group as Galois group and any prescribed commutative k-algebra R as invariant ring. This is a generalization of the Artin–Schreier–Witt theory of modular Galois field extensions of degree ps.
  • Fleischmann, P. and Woodcock, C. (2011). Relative Invariants, Ideal Classes and Quasi-Canonical Modules of Modular Rings of Invariants. Journal of Algebra [Online] 348:110-134. Available at: http://dx.doi.org/10.1016/j.jalgebra.2011.09.024.
    We describe “quasi-canonical modules” for modular invariant rings R of finite group actions on factorial Gorenstein domains. From this we derive a general “quasi-Gorenstein criterion” in terms of certain 1-cocycles. This generalizes a recent result of A. Braun for linear group actions on polynomial rings, which itself generalizes a classical result of Watanabe for non-modular invariant rings.

    We use an explicit classification of all reflexive rank one R-modules, which is given in terms of the class group of R, or in terms of R-semi-invariants. This result is implicitly contained in a paper of Nakajima (1982).
  • Ayad, M. and Fleischmann, P. (2008). On the decomposition of rational functions. Journal of Symbolic Computation [Online] 43:259-274. Available at: http://dx.doi.org/10.1016/j.jsc.2007.10.009.
    Let f := p/q epsilon K(x) be a rational function in one variable. By Luroth's theorem, the collection of intermediate fields K(f) subset of L subset of K(x) is in bijection with inequivalent proper decompositions f = g circle h, with g, h epsilon K(x) of degrees >= 2. In [Alonso, Cesar, Gutierrez, Jaime, Recio, Tomas, 1995. A rational function decomposition algorithm by near-separated polynomials. J. Symbolic Comput. 19, 527-544] an algorithm is presented to calculate such a function decomposition. In this paper we describe a simplification of this algorithm, avoiding expensive solutions of linear equations. A MAGMA implementation shows the efficiency of our method. We also prove some indecomposability criteria for rational functions, which were motivated by computational experiments.
  • Fleischmann, P., Kemper, G. and Woodcock, C. (2007). Homomorphisms, Localizations and a new Algorithm to construct Invariant Rings of Finite Groups. Journal of Algebra [Online] 309:497-517. Available at: http://dx.doi.org/10.1016/j.jalgebra.2005.06.038.
    Let G be a finite group acting on a polynomial ring A over the field K and let AG denote the corresponding ring of invariants. Let B be the subalgebra of AG generated by all homogeneous elements of degree less than or equal to the group order |G|. Then in general B is not equal to AG if the characteristic of K divides |G|. However we prove that the field of fractions Quot(B) coincides with the field of invariants Quot(AG)=Quot(A)G. We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for AG. We prove that there is always a nonzero transfer cAG of degree <|G|, such that the localization (AG)c can be generated by fractions of homogeneous invariants of degrees less than 2|G|?1. If with finite-dimensional -module V, then c can be chosen in degree one and 2|G|?1 can be replaced by |G|. Let denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that contains the transfer ideal and thus can be used to calculate generators for AG by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for AG.
  • Shank, R. et al. (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."
  • 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)}.
  • Fleischmann, P. (2004). On Invariant Theory of Finite Groups. CRM Proceedings and Lecture Notes 35:43-70.
  • 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., Holder, M. and Roelse, P. (2003). The Black-Box Niederreiter Algorithm and its Implementation Over the Binary Field. Mathematics of Computation [Online] 72:1887-1899. Available at: http://dx.doi.org/10.1090/S0025-5718-03-01494-7.
    The most time-consuming part of the Niederreiter algorithm for factoring univariate polynomials over finite fields is the computation of elements of the nullspace of a certain matrix. This paper describes the so-called "black-box" Niederreiter algorithm, in which these elements are found by using a method developed by Wiedemann. The main advantages over an approach based on Gaussian elimination are that the matrix does not have to be stored in memory and that the computational complexity of this approach is lower. The black-box Niederreiter algorithm for factoring polynomials over the binary field was implemented in the C programming language, and benchmarks for factoring high-degree polynomials over this field are presented. These benchmarks include timings for both a sequential implementation and a parallel implementation running on a small cluster of workstations. In addition, the Wan algorithm, which was recently introduced, is described, and connections between (implementation aspects of) Wan's and Niederreiter's algorithm are given.
  • Fleischmann, P. and Shank, R. (2003). The relative trace ideal and the depth of modular rings of invariants. Archiv der Mathematik 80:347-353.
    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.
  • Fleischmann, P. (2002). On pointwise conjugacy of distinguished coset representatives in Coxeter groups. Journal of Group Theory [Online] 5:269-283. Available at: http://dx.doi.org/10.1515/jgth.2002.002.
    Let (W, S) be a Coxeter system. For a standard parabolic. subgroup W-K,
    K subset of or equal to S let D-K be the set of distinguished coset
    representatives, i.e. representatives of cosets W(K)w of minimal
    Coxeter length. If L = K-c subset of or equal to S with c is an element
    of W, then D-K and D-L = c(-1) D-K are in general not conjugate as
    sets. However it is shown that if WK is finite, they are conjugate
    'pointwise', i.e. there is a bijection theta : D-K --> D-L such that
    theta(d) = d(wc) for some w is an element of W-K depending on d is an
    element of D-K. In particular for each conjugacy class C of W the
    cardinalities # (D-K boolean AND C) and # (D-L boolean AND C) are the
    same. The case of infinite standard parabolic subgroups is also
    discussed and a corresponding result is proved.
  • Fleischmann, P., Lempken, W. and Zalesskii, A. (2001). Linear groups over GF(2(k)) generated by a conjugacy class of a fixed point free element of order 3. Journal of Algebra [Online] 244:631-663. Available at: http://dx.doi.org/10.1006/jabr.2001.8852.
  • Fleischmann, P. (2000). The Noether bound in invariant theory of finite groups. Advances in Mathematics [Online] 156:23-32. Available at: http://dx.doi.org/10.1006/aima.2000.1952.
    Let R be a commutative ring, V a finitely generated free R-module and G less than or equal to GL(R)(V) a finite group acting naturally on the graded symmetric algebra A = Sym(V). Let beta (A(G)) denote the minimal number m, such that the ring A(G) of invariants can be generated by finitely many elements of degree at most m. Furthermore, let H <<vertical bar> G be a normal subgroup such that the index \G : H\ is invertible in R. In this paper we prove the inequality beta (A(G)) less than or equal to beta (A(H)) . \G : H\. For H = 1 and \G\ invertible in R we obtain Noether's bound beta (A(G)) less than or equal to \G\, which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order, \G\!, is invertible in R.
  • Paar, C., Fleischmann, P. and Soria-Rodriguez, P. (1999). Fast arithmetic for public-key algorithms in Galois fields with composite exponents. IEEE Transactions on Computers [Online] 48:1025-1034. Available at: http://dx.doi.org/10.1109/12.805153.
    This contribution describes a new class of arithmetic architectures for Galois fields GF(2(k)). The main applications of the architecture are public-key systems which are based on the discrete logarithm problem fdr elliptic curves. The architectures use a representation of the field GF(2(k)) as GF((2(n))(m)), where k = n.m. The approach explores bit parallel arithmetic in the subfield GF(2(n)) and serial processing for the extension field arithmetic. This mixed parallel-serial (hybrid) approach can lead to fast implementations. As the core module, a hybrid multiplier is introduced and several optimizations are discussed. We provide two different approaches to squaring. We develop exact expressions for the complexity of parallel squarers in composite fields, which can have a surprisingly low complexity. The hybrid architectures are capable of exploring the time-space trade-off paradigm in a flexible manner. In particular, the number of dock cycles for one field multiplication, which is the atomic operation in most public-key schemes, can be reduced by a factor of n compared to other known realizations. The acceleration is achieved at the cost of an increased computational complexity. We describe a proof-of-concept implementation of an ASIC for multiplication and squaring in GF((2(n))(m)), m variable.

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.
  • Elmer, J. and Fleischmann, P. (2010). On the Depth of Modular Invariant Rings for the Groups Cp x Cp. in: Campbell, E. et al. eds. Symmetry and Spaces: In Honor of Gerry Schwarz. Boston: Birkhäuser, pp. 45-61. Available at: http://dx.doi.org/10.1007/978-0-8176-4875-6.
    Let G be a finite group, k a field of characteristic p and V a finite dimen-
    sional kG-module. Let R :=Sym(V?), the symmetric algebra over the dual spaceV?,
    with G acting by graded algebra automorphisms. Then it is known that the depth of
    the invariant ring RG is at least min{dim(V),dim(VP)+ccG(R)+1}. A module V
    for which the depth of RG attains this lower bound was called flat by Fleischmann,
    Kemper and Shank [13]. In this paper some of the ideas in [13] are further developed
    and applied to certain representations of Cp ×Cp, generating many new examples
    of flat modules. We introduce the useful notion of “strongly flat” modules, classi-
    fying them for the group C2 ×C2, as well as determining the depth of RG for any
    indecomposable modular representation of C2×C2.

Conference or workshop item

  • Fleischmann, P. (1997). Relative trace ideals and Cohen-Macaulay quotients of modular invariant rings. in: Draxler, P., Michler, G. O. and Ringel, C. M. eds. Euroconference on Computational Methods for Representations of Groups and Algebras. Germany: Birkhauser Verlag, pp. 211-233.
    Let G be a finite group, IF a field whose characteristic p divides the order of G and AG the invariant ring of a finite-dimensional FG-module V. In analogy to modular representation theory we define for any subgroup H less than or equal to G the (relative) trace-ideal A(H)(G) del a A(G) to be the image of the relative trace map t(G)(H) : A(H) --> A(G), f bar right arrow Sigma(g is an element of[G:H]) g(f). Moreover, for any family chi of subgroups of G, we define the relative trace ideals A behaviour. If chi consists of proper subgroups of a Sylow p-group P less than or equal to G, then A(chi)(G) is always a proper ideal of AG; in fact, we show that its height is bounded above by the codimension of the fixed point space VP. But We also prove that if V is relatively X-projective, then A(chi)(G) still contains all invariants of degree not divisible by p. If V is projective then this result applies in particular to the (absolute) trace ideal A({e})(G). We also give a [geometric analysis' of trace ideals, in particular of the ideal A(<P)(G) := Sigma(Q<P) A(Q)(G)0, and show that I-G,I-P := root(A<P)(G) is a prime ideal which has the geometric interpretation as 'vanishing ideal' of G-orbits with length coprime to p. It is shown that A(G)/I-G,I-P is always a Cohen-Macaulay algebra, if the action of P is defined over the prime field. This generalizes a well known result of Hochster and Eagon for the case. P = 1 (see [13]). Moreover we prove that if V is a direct summand of a permutation module (i.e. a 'trivial source module'), then the A(chi)(G) are radical ideals and A(<P)(G) = I-G,I-P. Hence in this case the ideal and the corresponding Cohen-Macaulay quotient can be constructed using relative trace maps.
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