Finite Symplectic Groups Research Articles

Appendix: Non-uniqueness of supercuspidal support for finite reductive groups

We exhibit a simple representation of the finite symplectic group of Sp8(q) in characteristic ℓ|q2+1 having two non-conjugate supercuspidal supports. This answers a question of M.-F. Vignéras.

Enumeration of Commuting Pairs in Lie Algebras over Finite Fields

Feit and Fine derived a generating function for the number of ordered pairs of commuting $$n{\times}n$$ matrices over the finite field $${\mathbb{F}_{q}}$$ . This has been reproved and studied by Bryan and Morrison from the viewpoint of motivic Donaldson-Thomas theory. In this note, we give a new proof of the Feit-Fine result, and generalize it to the Lie algebra of finite unitary groups and to the Lie algebra of odd characteristic finite symplectic groups. We extract some asymptotic information from these generating functions. Finally, we derive generating functions for the number of commuting nilpotent elements for the Lie algebras of the finite general linear and unitary groups, and of odd characteristic symplectic groups.

Real Representations of Finite Symplectic Groups over Fields of Characteristic Two

Abstract We prove that when q is a power of 2 every complex irreducible representation of $\textrm{Sp}\big (2n, \mathbb{F}_{q}\big )$ may be defined over the real numbers, that is, all Frobenius–Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of $\textrm{Sp}\big(2n, \mathbb{F}_{q}\big )$, or of $\textrm{SO}\big(2n+1,\mathbb{F}_{q}\big )$, for any prime power q.

Action of automorphisms on irreducible characters of symplectic groups

Assume G is a finite symplectic group Sp2n(q) over a finite field Fq of odd characteristic. We describe the action of the automorphism group Aut(G) on the set Irr(G) of ordinary irreducible characters of G. This description relies on the equivariance of Deligne–Lusztig induction with respect to automorphisms. We state a version of this equivariance which gives a precise way to compute the automorphism on the corresponding Levi subgroup; this may be of independent interest. As an application we prove that the global condition in Späth's criterion for the inductive McKay condition holds for the irreducible characters of Sp2n(q).

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. III

Following our papers [1] and [2] (Parts I and II), we classify degenerations of codimension 2 or more of Kählerian surfaces with finite symplectic automorphism groups. In [1] and [2] this was done for codimension .

Lattices generated by joins of the flats in orbits under finite affine-singular symplectic group and its characteristic polynomials

Let ASG(2ν + l, ν;F q ) be the (2ν + l)-dimensional affine-singular symplectic space over the finite field F q and ASp2ν+l,ν (F q ) be the affine-singular symplectic group of degree 2ν + l over F q . Let O be any orbit of flats under ASp2ν+l,ν (F q ). Denote by L J the set of all flats which are joins of flats in O such that O ⊆ L J and assume the join of the empty set of flats in ASG(2ν + l, ν;F q ) is ∅. Ordering L J by ordinary or reverse inclusion, then two lattices are obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice L J , when the lattices form geometric lattice, lastly gives the characteristic polynomial of L J .

Inductive McKay condition for finite simple groups of type 𝖢

We verify the inductive McKay condition for simple groups of Lie type C \mathsf {C} , namely finite projective symplectic groups. This contributes to the program of a complete proof of the McKay conjecture for all finite groups via the reduction theorem of Isaacs-Malle-Navarro and the classification of finite simple groups. In an important step we use a new counting argument to determine the stabilizers of irreducible characters of a finite symplectic group in its outer automorphism group. This is completed by analogous results on characters of normalizers of Sylow d d -tori in those groups.

On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g\rangle$ for every element $g \in G$. In [Sib. Math. J. 2015. Vol. 56, no. 6] we proved that subgroups of odd indeces are pronormal in many finite simple group. In [Proc. Steklov Inst. Math., to appear] we proved that a group $PSp_{6n}(q)$ with $q \equiv \pm 3 \pmod 8$ contains a nonpronormal subgroup of odd index. In this paper we prove that if $n \not \in \{2^m, 2^m(2^{2k}+1) \mid m, k \in \mathbb{N} \cup \{0\}\}$ then a group $PSp_{2n}(q)$ with $q \equiv \pm 3 \pmod 8$ contains a nonpronormal subgroup of odd index; if $n=2^m$ then any subgroup of odd index is pronormal in a group $PSp_{2n}(q)$, where $q \equiv \pm 3 \pmod 8$.

Asymptotics of the number of involutions in finite classical groups

Abstract Answering a question of Geoff Robinson, we compute the large n limiting proportion of i GL ⁢ ( n , q ) / q ⌊ n 2 / 2 ⌋ {i_{\mathrm{GL}}(n,q)/q^{\lfloor n^{2}/2\rfloor}} , where i GL ⁢ ( n , q ) {i_{\mathrm{GL}}(n,q)} denotes the number of involutions in the group GL ⁢ ( n , q ) {\mathrm{GL}(n,q)} . We give similar results for the finite unitary, symplectic, and orthogonal groups, in both odd and even characteristic. At the heart of this work are certain new “sum = product” identities. Our self-contained treatment of the enumeration of involutions in even characteristic symplectic and orthogonal groups may also be of interest.

On spectra of almost simple groups with symplectic or orthogonal socle

Finite groups are said to be isospectral if they have the same sets of the orders of elements. We investigate almost simple groups H with socle S, where S is a finite simple symplectic or orthogonal group over a field of odd characteristic. We prove that if H is isospectral to S, then H/S presents a 2-group. Also we give a criterion for isospectrality of H and S in the case when S is either symplectic or orthogonal of odd dimension.

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. II

We prove the main conjecture (Conjecture 7.1) of the paper [1]. Using this result, we classify degenerations of codimension 1 of Kählerian K3 surfaces with finite symplectic automorphism groups.

Weil representations of finite symplectic groups and finite odd-dimensional orthogonal groups

In [8], Srinivasan decomposes the uniform projection of the character of Weil representation ωG,G′ of a finite reductive dual pair (G,G′) in terms of Deligne–Lusztig virtual characters when the dual pair is one of the following: (1) two general linear groups; (2) two unitary groups; (3) a symplectic group and an even-dimensional orthogonal group, assuming the order of the finite field is large enough. In this article we obtain the decomposition for the remaining case, i.e., the dual pair of a symplectic group and an odd-dimensional orthogonal group. In particular, we have the decomposition of the uniform projection of the character of the Weil representation of a finite symplectic group.

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. II

Доказывается основная гипотеза 7.1 статьи [1]. С помощью этого результата классифицируются вырождения коразмерности 1 кэлеровых K3-поверхностей с конечными симплектическими группами автоморфизмов. Библиография: 20 наименований.

Recognition by spectrum for simple classical groups in characteristic 2

A finite group G is said to be recognizable by spectrum if every finite group with the same set of element orders as G is isomorphic to G. We prove that all finite simple symplectic and orthogonal groups over fields of characteristic 2, except S4(q), S6(2), O8+ (2), and S8(q), are recognizable by spectrum. This result completes the study of the recognition-by-spectrum problem for finite simple classical groups in characteristic 2.

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups

Using the results of [1]–[3] on Kählerian K3 surfaces and Niemeier lattices, we classify degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups with emphasis on degenerations of codimension 1.

On the structure of finite groups isospectral to finite simple groups

Abstract Finite groups are said to be isospectral if they have the same sets of element orders. A finite nonabelian simple group L is said to be almost recognizable by spectrum if every finite group isospectral to L is an almost simple group with socle isomorphic to L. It is known that all finite simple sporadic, alternating and exceptional groups of Lie type, except J 2, A 6, A 10 and 3 D 4(2), are almost recognizable by spectrum. The present paper is the final step in the proof of the following conjecture due to V. D. Mazurov: there exists a positive integer d 0 such that every finite simple classical group of dimension larger than d 0 is almost recognizable by spectrum. Namely, we prove that a nonabelian composition factor of a finite group isospectral to a finite simple symplectic or orthogonal group L of dimension at least 10, is either isomorphic to L or not a group of Lie type in the same characteristic as L, and combining this result with earlier work, we deduce that Mazurov's conjecture holds with d 0 = 60.

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups

При использовании результатов из [1]-[3], где изучались кэлеровы К3-поверхности и решетки Нимейера, классифицируются вырождения кэлеровых К3-поверхностей с конечными симплектическими группами автоморфизмов. Главным образом, классифицируются вырождения коразмерности 1. Библиография: 23 наименования.

The Maximum Order of the Elements of a Finite Symplectic Group of Even Characteristic

We give an exact formula, as a function of m and q, for the maximum order of the elements of the finite symplectic group Sp2m (q), with q even, and of its automorphism group.

Strongly real classes in finite unitary groups of odd characteristic

Abstract. We classify all strongly real conjugacy classes of the finite unitary group U(n,𝔽 q ) when q is odd. In particular, we show that g ∈ U(n,𝔽 q ) is strongly real if and only if g is an element of some embedded orthogonal group O±(n,𝔽 q ). Equivalently, g is strongly real in U(n,𝔽 q ) if and only if g is real and every elementary divisor of g of the form (t ± 1)2m has even multiplicity. We apply this to obtain partial results on strongly real classes in the finite symplectic group Sp(2n,𝔽 q ), q odd, and a generating function for the number of strongly real classes in U(n,𝔽 q ), q odd, and we also give partial results on strongly real classes in U(n,𝔽 q ) when q is even.

Normalized Matching Property of Posets Generated by Orbits of Subspaces Under Finite Symplectic Groups

Let be the 2ν-dimensional symplectic space over a finite field 𝔽 q , and let ℳ be a given nontrivial orbit of subspaces in under the symplectic group Sp 2ν(𝔽 q ). Denote by 𝒫 the set of subspaces which are intersections of subspaces in ℳ. By ordering 𝒫 by inclusion, 𝒫 is a finite graded poset. In this article we show that 𝒫 has the normalized matching (NM) property, which implies that 𝒫 has the strong Sperner property and the Lubell-Yamamoto-Meschalkin (LYM) property.