Piotr Mizerka
We compute the characters of real irreducible representations of SL(2,q), the special linear group on q letters, for an odd prime $q$. Moreover, we give the dimensions of these irreducible representations under the actions of cyclic subgroups of SL(2,q).
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Suppose that $G$ is a finite group and $H$ is a nilpotent subgroup of $G$. If a character of $H$ induces an irreducible character of $G$, then the generalized Fitting subgroup of $G$ is nilpotent.
Let G be a finite group with cyclic Sylow p-subgroup, and let k be a field of characteristic p. Then H^*(BG;k) and H_*(ΩBG\phat;k) are A_{\infty} algebras whose structure we determine up to quasi-isomorphism.
Let $n\geq 3,$ and let $Y$ be a simply connected, simple algebraic group of type $D_{n+1}$ over an algebraically closed field $K.$ Also let $X$ be the subgroup of type $B_n$ of $Y,$ embedded in the usual way. In this paper, we correct an error in a proof of a theorem of Seitz, resulting in the discovery of a new family of triples $(X,Y,V),$ where $V$ denotes a finite-dimensional, irreducible, rational $KY$-module, on which $X$ acts irreducibly. We go on to investigate the impact of the existence of the new examples on the classification of the maximal closed connected subgroups of the classica…
In this paper, we obtain explicit branching laws for all irreducible unitary representations of $\Spin(N,1)$ restricted to a parabolic subgroup $P$. The restriction turns out to be a finite direct sum of irreducible unitary representations of $P$. We also verify Duflo's conjecture for the branching law of tempered representations of $\Spin(N,1)$ with respect to a parabolic subgroup $P$. That is to show: in the framework of the orbit method, the branching law of a tempered representation is determined by the behavior of the moment map from the corresponding coadjoint orbit. A few key tools used…
The alpha-determinant unifies and interpolates the notion of the determinant and permanent. We determine the irreducible decomposition of the cyclic module of $gl_n(C)$ defined by the alpha-determinant. The degeneracy of the irreducible decomposition is determined by the content polynomial of a given partition.
Let $G$ be a finite group with a Sylow $p$-subgroup $P$. We prove that the principal $p$-blocks of $G$ and $N_G(P)$ are perfectly isometric under the assumption $G$ has a cyclic $p$-hyperfocal subgroup.
We give a detailed account of a combinatorial construction, due to Cherednik, of cyclic generators for irreducible modules of the affine Hecke algebra of the general linear group with generic parameter q.
Let $G$ be a finite group, $H$ be a normal subgroup of prime index $p$. Let $F$ be a field of either characteristic $0$ or prime to $|G|$. Let $η$ be an irreducible $F$-representation of $H$. If $F$ is an algebraically closed field of characteristic either $0$ or prime to $|G|$, then the induced representation $η\uparrow^{G}_{H}$ is either irreducible or a direct sum of $p$ pairwise inequivalent irreducible representations. In this paper, we show that if $F$ is not assumed algebraically closed field, then there are five possibilities in the decomposition of induced representation into irreduci…
Let $H$ be a linear algebraic group whose connected component $G\neq 1$ is simple and $H/G$ is cyclic. We determine the irreducible projective representations $ϕ$ of $H$ such that $ϕ(G)$ is irreducible and $ϕ(h)$ has simple spectrum for some $h\in H$. The latter means that all irreducible constituents of the group $ϕ( \langle h\rangle)$ are of multiplicity 1. (Here $\langle h\rangle$ is the subgroup of $H$ generated by $h$.) This extends an earlier known result for $H=G$.
In this notes we construct and count all ordinary irreducible characters of Sylow $p$-subgroups of the Steinberg triality groups ${}^3D_4(p^{3m})$.
The exact degree bound for the generators of rings of polynomial invariants is determined for the finite, non-cyclic groups having a cyclic subgroup of index two. It is proved that the Noether number of these groups equals one half the order of the group plus 1 or 2.
Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$. Let ${\mathcal U}_n(A)$ be the subgroup of $\mathrm{GL}_n(A)$ of all upper triangular matrices with 1's along the main diagonal. Let $P=H_n(A)\rtimes {\mathcal U}_n(A)$, where ${\mathcal U}_n(A)$ acts on $H_n(A)$ by $*$-congruence transformations. We may view $P$ as a unipotent subgroup of either a symplectic group $\mathrm{Sp}_{2n}(A)$, if $*=1_A$ (in which c…
For a finite group $G$ and complex character $χ\in\mathrm{Irr}(G)$ that restricts irreducibly to a normal subgroup $N\vartriangleleft G,$ we prove a theorem about Clifford correspondences between the characters of subgroups of $G$ that induce $χ.$
In this note, we formulate an observation that "almost all" irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.
We prove that non-trivial representations of the alternating group $A_n$ are reducible over a primitive proper subgroup which is isomorphic to some alternating group $A_m$.
Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module. We say that $(G,H,V)$ is an irreducible triple if $V$ is irreducible as a $KH$-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups…
Let $K$ be an abelian extension of the rationals. Let $S(K)$ be the Schur group of $K$ and let $CC(K)$ be the subgroup of $S(K)$ generated by classes containing cyclic cyclotomic algebras. We characterize when $CC(K)$ has finite index in $S(K)$ in terms of the relative position of $K$ in the lattice of cyclotomic extensions of the rationals.
An irreducible decomposition of the Weil representation restricted to open compact subgroups
本文利用 Schrödinger 模型，对任意剩余特征的 p-进域，给出了 Weil 表示限制到极大开紧子群和 Iwahori 子群的不可约分解，并证明其无重数性。
Minimal polynomials of $p$-elements of finite groups of Lie type with cyclic Sylow $p$-subgroups
该研究将有限李型群 $p$-元素极小多项式的既有跨特征表示结果，推广到 Sylow $p$-子群循环且 $p$ 异于表示域特征的情形。
We extend the Howlett-Isaacs theorem on the solvability of groups of central type taking into account actions by automorphisms. Then we study certain induced characters whose constituents have all the same degree.
We characterize finite groups having a cyclic Sylow p-subgroup in terms of the action of a specific Galois automorphism on the principal p-block for p=2,3. We show that the analog statement for blocks with arbitrary defect group would follow from the blockwise McKay-Navarro conjecture.
Let $G$ be a simple classical algebraic group over an algebraically closed field $K$ of characteristic $p \ge 0$ with natural module $W$. Let $H$ be a closed subgroup of $G$ and let $V$ be a non-trivial irreducible tensor-indecomposable $p$-restricted rational $KG$-module such that the restriction of $V$ to $H$ is irreducible. In this paper we classify all such triples $(G,H,V)$, where $H$ is a maximal closed disconnected positive-dimensional subgroup of $G$, and $H$ preserves a natural geometric structure on $W$.
We classify all triples $(G,V,H)$ such that $SL_n(q)\leq G\leq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $\FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $V\dar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.
Let $G_2(q)$ be the Chevalley group of type $G_2$ defined over a finite field with q=p^n elements, where p is a prime number and $n$ is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible representation of $G$ in characteristic other than p to a maximal subgroup of $G_2(q)$ is still irreducible. Similar results are obtained for $^2B_2(q)$ and $^2G_2(q)$.
Standard Generators of Finite Fields and their Cyclic Subgroups
本文给出有限域及其乘法循环子群的标准化构造与生成元，旨在为各类软件和数学数据集提供 Conway 多项式的可替代方案。
Let G be a simple classical algebraic group over an algebraically closed field K of characteristic $p \ge 0$ with natural module W. Let H be a closed subgroup of G and let V be a nontrivial p-restricted irreducible tensor indecomposable rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V) of this form, where $V \ne W, W^{*}$ and H is a disconnected almost simple positive-dimensional closed subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classification of the irreducible …
We construct a differential graded algebra (DGA) modelling certain $A_\infty$ algebras associated with a finite group $G$ with cyclic Sylow subgroups, namely $H^*BG$ and $H_*ΩBG^{^\wedge}_p$. We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander--Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups.
The classical Itô-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group $G$ is coprime to a given prime $p$, then $G$ has a normal Sylow $p$-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of $G$ is less than $4/3$ then $G$ has a normal Sylow $2$-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These res…
Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.
Let E be a finite extension of Fp. Using Fontaine's theory of (phi,Gamma)-modules, Colmez has shown how to attach to any irreducible E-linear representation of Gal(Qpbar/Qp) an infinite dimensional smooth irreducible E-linear representation of B_2(Qp) that has a central character. We prove that every such representation of B_2(Qp) arises in this way. Our proof extends to algebraically closed fields E of characteristic p. In this case, infinite dimensional smooth irreducible E-linear representations of B_2(Qp) having a central character arise in a similar way from irreducible E-linear represent…
Let $G$ be a simple linear algebraic group defined over an algebraically closed field of characteristic $p\geq 0$ and let $ϕ$ be a $p$-restricted irreducible representation of $G$. Let $T$ be a maximal torus of $G$ and $s\in T$. We say that $s$ is strongly regular if $α(s)\neβ(s)$ for all distinct $T$-roots $α$ and $β$ of $G$. Our main result states that if all but one of the eigenvalues of $ϕ(s)$ are of multiplicity 1 then, with a few specified exceptions, $s$ is strongly regular. This can be viewed as an extension of our earlier result saying that under the same hypotheses, $s$ must be regul…
Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p\geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial irreducible $KG$-module, which is $p$-restricted, tensor indecomposable and rational. Assume that the restriction of $V$ to $H$ is irreducible. In this paper, we study the triples $(G,H,V)$ of this form when $G$ is a classical group and $H$ is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples $(G,H,V)$ to the case w…
The purpose of the present paper is to investigate a hypergroup arising from irreducible characters of a compact group G and a closed subgroup of G with finite index. The convolution of this hypergroup is introduced by inducing irreducible representations and by restricting irreducible representations. The method of proof relies on character formulae of induced representations of compact groups and of Frobenius' reciprocity theorem.
Let $U(q)$ be a Sylow $p$-subgroup of the Chevalley groups $D_4(q)$ where $q$ is a power of a prime $p$. We describe a construction of all complex irreducible characters of $U(q)$ and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in $q-1$ with nonnegative coefficients.
We study irreducible restrictions from modules over symmetric groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. Such results are known when the characteristic of the ground field is greater than $3$, but the small characteristics cases require a substantially more delicate analysis and new ideas. This work fits into the Aschbacher-Scott program on maximal subgroups of finite classical groups.
We parametrize the set of irreducible characters of the Sylow $p$-subgroups of the Chevalley groups $\mathrm{D}_6(q)$ and $\mathrm{E}_6(q)$, for an arbitrary power $q$ of any prime $p$. In particular, we establish that the parametrization is uniform for $p \ge 3$ in type $\mathrm{D}_6$ and for $p \ge 5$ in type $\mathrm{E}_6$, while the prime $2$ in type $\mathrm{D}_6$ and the primes $2,$ $3$ in type $\mathrm{E}_6$ yield character degrees of the form $q^m/p^i$ which force a departure from the generic situations. Also for the first time in our analysis we see a family of irreducible characters …
We describe the centralizer of irreducible representations from a finitely generated group $Γ$ to $PSL(p,\mathbb{C})$ where $p$ is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety $χ^i(Γ,PSL(p,\mathbb{C}))$. When $Γ$ is a free group of rank $l\geq 2$ or the fundamental group of a closed Riemann surface of genus $g\geq 2$, we give a complete description of this locus and prove that this locus is exactly the set of…
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