Search

We develop a theory of modular forms on the groups SO ( 3 , n + 1 ) , n ≥ 3 . This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of SO ( 3 , n + 1 ) , there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on SO ( 2 , n ) . As an application, using a local result of Savin, we prove that the so-called “next-to-minimal” modular form on quaternionic E 8 has rational Fourier expansion.

In this paper, we give a precise definition of an analytic γ-factor of an irreducible representation of a classical group over a local function field of odd characteristic so that it satisfies some notable properties which are enough to define it uniquely. We use the doubling method to define the γ-factor, and the main theorem extends works of Lapid-Rallis, Gan, Yamana, and the author to a classical group over a local function field of odd characteristic.

Let [Formula: see text] be a classical group over an arbitrary field [Formula: see text], acting on an [Formula: see text]-dimensional [Formula: see text]-space [Formula: see text]. All those maximal subgroups of [Formula: see text] are classified each of which normalizes a solvable subgroup [Formula: see text] of [Formula: see text] not lying in [Formula: see text].

We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow $p$-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic $p$-group acting on an elementary abelian $p$-group., Comment: Example 5.12 has been corrected and expanded

Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of [Formula: see text]-unit groups of orders. The algorithm is then used for some investigations concerning the congruence subgroup property.

This paper generalizes the GGP conjectures which were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the nongeneric L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of A-parameters which governs the branching laws for $GL_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in our earlier work including Bessel models and Fourier-Jacobi models., 70 pages, to appear in Compositio Math

We give a short combinatorial proof of Asai's decomposition formula for Lusztig induction of unipotent characters in groups of classical type, relying solely on the Mackey formula.

A GLd-pseudocharacter is a function from a group Γ to a ring k satisfying polynomial relations that make it “look like” the character of a representation. When k is an algebraically closed field of characteristic 0, Taylor proved that GLd-pseudocharacters of Γ are the same as degree-d characters of Γ with values in k, hence are in bijection with equivalence classes of semisimple representations Γ → GLd(k). Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group H over an algebraically closed field k of characteristic 0 and for any group Γ, there exists an infinite collection of functions and relations which are naturally in bijection with H(k)-conjugacy classes of semisimple homomorphisms Γ→ H(k). In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG algebra. We then define generating sets and generating relations for these objects and show that, for all H as above, the corresponding FFG-algebra is finitely presented up to radical. Hence one can always define H-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations up to radical of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of homomorphisms, following Larsen.

We analyse the complexity of constructing involution centralisers in unitary groups over fields of odd order. In particular, we prove logarithmic bounds on the number of random elements required to generate a subgroup of the centraliser of a strong involution that contains the last term of its derived series. We use this to strengthen previous bounds on the complexity of recognition algorithms for unitary groups in odd characteristic. Our approach generalises and extends two previous papers by the second author and collaborators on strong involutions and regular semisimple elements of linear groups., Comment: 48 pages. Revised after suggestions from Eamonn O'Brien and an anonymous referee. To appear J. Algebra

For each family of finite classical groups, and their associated simple quotients, we provide an explicit presentation on a specific generating set of size at most 8. Since there exist efficient algorithms to construct this generating set in any copy of the group, our presentations can be used to verify claimed isomorphisms between representations of the classical group. The presentations are available in Magma .

We introduceadmissible collectionsfor a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy theQuillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of|G|\lvert G\rvert. Compared to the methods in [M. Aschbacher and S. D. Smith, On Quillen’s conjecture for the 𝑝-groups complex,Ann. of Math. (2)137(1993), 3, 473–529], our techniques are simpler.

Recently, there has been substantial progress on the Alperin weight conjecture. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for simple groups of classical type under some additional assumption.

We parametrize the set of irreducible characters of the Sylow p-subgroups of the Chevalley groups D 6 ( q ) and E 6 ( q ) , for an arbitrary power q of any prime p. In particular, we establish that the parametrization is uniform for p ≥ 3 in type D 6 and for p ≥ 5 in type E 6 , while the prime 2 in type D 6 and the primes 2, 3 in type 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 of a classical group of degree q m / p i where i > 1 which occurs in type D 6 .

In this paper we show that when $\mathrm{G}$ is a classical semi-simple algebraic group, $\mathrm{B}\subset\mathrm{G}$ a Borel subgroup, and $\mathrm{X} = \mathrm{G}/\mathrm{B}$, then the structure coefficients of the Belkale-Kumar product $\odot_{0}$ on $\mathrm{H}^{*}(\mathrm{X}, \mathbf{Z})$ are all either $0$ or $1$.

Let $G$ be a classical group with natural module $V$ and Lie algebra $\mathfrak{g}$ over an algebraically closed field $K$ of good characteristic. For rational irreducible representations $f: G \rightarrow \operatorname{GL}(W)$ occurring as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, we describe the Jordan normal form of $\mathrm{d} f(e)$ for all nilpotent elements $e \in \mathfrak{g}$. The description is given in terms of the Jordan block sizes of the action of $e$ on $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$, for which recursive formulae are known. Our results are in analogue to earlier work (Proc. Amer. Math. Soc., 147 (2019) 4205-4219), where we considered these same representations and described the Jordan normal form of $f(u)$ for every unipotent element $u \in G$., to appear in J. Pure Appl. Algebra

On a product manifold (M, r), we consider four geometric structures compatible with r, e.g. hyper-paracomplex or bi-Lagrangian, and define distinguished generalized complex or paracomplex structures on M, which interpolate between some pairs of them. We study the twistor bundles whose smooth sections are these new structures, obtaining the typical fibers as homogeneous spaces of classical groups. Also, we give examples of product manifolds admitting some of these new structures.

Let A be the ring of adeles of a number field F. Given a self-dual irreducible, automorphic, cuspidal representation τ of GL n ( A ) , with a trivial central character, we construct its full inverse image under the weak Langlands functorial lift from the appropriate split classical group G. We do this by a new automorphic descent method, namely the double descent. This method is derived from the recent generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan [CFGK17] , which represent the standard L-functions for G × GL n . Our results are valid also for double covers of symplectic groups.

Let [math] be a vector space of dimension [math] over [math] , a finite field of [math] elements, and let [math] be a linear group. A base for [math] is a set of vectors whose pointwise stabilizer in [math] is trivial. We prove that if [math] is a quasisimple group (i.e., [math] is perfect and [math] is simple) acting irreducibly on [math] , then excluding two natural families, [math] has a base of size at most 6. The two families consist of alternating groups [math] acting on the natural module of dimension [math] or [math] , and classical groups with natural module of dimension [math] over subfields of [math] .

We study residue formulas for push-forward in the equivariant K-theory of homogeneous spaces. For the classical Grassmannian, the residue formula can be obtained from the cohomological formula by a substitution. We also give another proof using symplectic reduction and the method involving the localization theorem of Jeffrey–Kirwan. We review formulas for other classical groups, and we derive them from the formulas for the classical Grassmannian. Next, we consider the homogeneous spaces for the exceptional group $$\mathbf{G}_2$$ . One of them, $$\mathbf{G}_2/P_2$$ corresponding to the shorter root, embeds in the Grassmannian Gr(2, 7). We find its fundamental class in the equivariant K-theory $$K^\mathbb {T}(Gr(2,7))$$ . This allows to derive a residue formula for the push-forward. It has significantly different character compared to the classical groups case. The factor involving the fundamental class of $$\mathbf{G}_2/P_2$$ depends on the equivariant variables. To perform computations more efficiently, we apply the basis of K-theory consisting of Grothendieck polynomials. The residue formula for push-forward remains valid for the homogeneous space $$\mathbf{G}_2/B$$ as well.

In this paper, we give an overview of combinatorial group testing and algebra. Our survey focuses on the constructions with algebraic methods, especially geometry of classical groups over finite fields.

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let G be an algebraic group of classical type with defining characteristic p > 0 , μ a dominant weight and W the Weyl group of G. Let G = G ( q ) be a finite classical group, where q is a p-power. For a weight μ of G the sum s μ of distinct weights w ( μ ) with w ∈ W viewed as a function on the semisimple elements of G is known to be a generalized Brauer character of G called an orbit character of G. We compute, for certain orbit characters and every maximal torus T of G, the multiplicity of the trivial character 1 T of T in s μ . The main case is where μ = ( q − 1 ) ω and ω is a fundamental weight of G. Let St denote the Steinberg character of G. Then we determine the unipotent characters occurring as constituents of s μ ⋅ S t defined to be 0 at the p-singular elements of G. Let β μ denote the Brauer character of a representation of S L n ( q ) arising from an irreducible representation of G with highest weight μ. Then we determine the unipotent constituents of the characters β μ ⋅ S t for μ = ( q − 1 ) ω , and also for some other μ (called strongly q-restricted). In addition, for strongly restricted weights μ, we compute the multiplicity of 1 T in the restriction β μ | T for every maximal torus T of G.

Let t be an involution in GL ( n , q ) whose fixed point space E + has dimension k between n / 3 and 2 n / 3 . For each g ∈ GL ( n , q ) such that t t g has even order, 〈 t t g 〉 contains a unique involution z ( g ) which commutes with t. We prove that, with probability at least c / log ⁡ n (for some c > 0 ), the restriction z ( g ) | E + is an involution on E + with fixed point space of dimension between k / 3 and 2 k / 3 . This result has implications in the analysis of the complexity of recognition algorithms for finite classical groups in odd characteristic. We discuss how similar results for involutions in other finite classical groups would solve a major open problem in our understanding of the complexity of constructing involution centralisers in those groups.

This paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix. (A square matrix $M$ is called almost cyclic if it is similar to a block-diagonal matrix with two blocks, such that one block is scalar and another block is a matrix whose minimum and characteristic polynomials coincide. Reflections and transvections are examples of almost cyclic matrices. The paper focuses on the Weil representations of finite classical groups, as there is strong evidence that these representations play a key role in the general picture., 45 pages

The double Pieri algebra, constructed by Howe, Lee, and the author in [10] , [12] , encodes information on the decomposition of Pieri type tensor products of irreducible representations for complex classical groups. In this paper we study its finite presentation in terms of generators and relations, and then prove that it admits the structure of a cluster algebra. We also study its sl 2 -module structure.

Symmetric k-varieties generalize Riemannian symmetric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric k-varieties are in one-to-one cor...

‎In the classical group theory there is‎ an open question‎: ‎Is every torsion free n-Engel group (for n ≥ 4)‎, nilpotent?‎. ‎To answer the question‎, ‎Traustason‎ [11] showed that with some additional conditions all‎ ‎4-Engel groups are locally nilpotent‎. ‎Here‎, ‎we gave some partial‎ answer to this question on Engel fuzzy subgroups‎. ‎We show that if μ is a normal 4-Engel fuzzy‎ subgroup of group G‎, ‎x,y in G and a =yx‎, ‎then μ| is a generalized nilpotent of class at‎ most 2‎. ‎Also we define a torsion free fuzzy subgroup and show‎ ‎that if μ is a 4-Engel torsion free fuzzy subgroup of G‎, ‎then μ| is a generalized nilpotent of class at most 4‎, ‎for conjugate elements a,y in G‎.

A relationship between the general linear group of degree n over a finite field and the integer partitions of n into parts of k different magnitudes was investigated recently by the author. In this paper, we use a variation of the classical binomial transform to derive a new connection between partitions into parts of k different magnitudes and another finite classical group, namely the symplectic group Sp. New identities involving the number of partitions of n into parts of k different magnitudes are introduced in this context.

Let G be a finite group of Lie type and St k be the Steinberg representation of G, defined over a field k. We are interested in the case where k has prime characteristic l and St k is reducible. Tinberg has shown that the socle of St k is always simple. We give a new proof of this result in terms of the Hecke algebra of G with respect to a Borel subgroup and show how to identify the simple socle of St k among the principal series representations of G. Furthermore, we determine the composition length of St k when G = GL n ( q ) or G is a finite classical group and l is a so-called linear prime.

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.

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements invariably generate $S_n$ is bounded away from zero by an absolute constant for all $n$. Subsequently, Eberhard, Ford and Green have shown that the probability that three randomly selected elements invariably generate $S_n$ tends to zero as $n \rightarrow \infty$. In this paper, we prove an analogous result for the finite classical groups. More precisely, let $G_r(q)$ be a finite classical group of rank $r$ over $\mathbb{F}_q$. We show that for $q$ large enough, the probability that four randomly selected elements invariably generate $G_r(q)$ is bounded away from zero by an absolute constant for all $r$, and for three elements the probability tends to zero as $q \rightarrow \infty$ and $r \rightarrow \infty$. We use the fact that most elements in $G_r(q)$ are separable and the well-known correspondence between classes of maximal tori containing separable elements in classical groups and conjugacy classes in their Weyl groups., 22 pages

The paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group $G$ of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group $G$ in terms of its Lie rank and defining characteristic. When $G$ is specified as $G(q)$ and $q$ is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of $G$., 34 pages (including 1 blank page created by arXiv processing); improved exposition

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic two. Any non-trivial self-dual irreducible $K[G]$-module $W$ admits a non-degenerate $G$-invariant alternating bilinear form, thus giving a representation $f: G \rightarrow \operatorname{Sp}(W)$. In the case where $G = \operatorname{SL}_n(K)$ and $W$ has highest weight $\varpi_1 + \varpi_{n-1}$, and in the case where $G = \operatorname{Sp}_{2n}(K)$ and $W$ has highest weight $\varpi_2$, we determine for every unipotent element $u \in G$ the conjugacy class of $f(u)$ in $\operatorname{Sp}(W)$. As a part of this result, we describe the conjugacy classes of unipotent elements of $\operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)$ in $\operatorname{Sp}(V_1 \otimes V_2)$., Comment: to appear in J. Algebra

In this paper, we begin with the classification of Harish-Chandra imprimitive representations in non-defining characteristic. We recall the connection of this problem to certain generalizations of Iwahori-Hecke algebras and show that Harish-Chandra induction is compatible with the Morita equivalence by Bonnaf\'{e} and Rouquier, thus reducing the classification problem to quasi-isolated blocks. Afterwards, we consider imprimitivity of unipotent representations of certain classical groups. In the case of general linear and unitary groups, our reduction methods then lead to results for arbitrary Lusztig series., Comment: Corrected several typos and misleading formulations

If a finite quasisimple group G with simple quotient S is embedded into a suitable classical group X through the smallest degree of a projective representation of S, then the normalizer of G in X is a maximal subgroup of X, up to two series of exceptions where S is a Ree group, and four exceptions where S is sporadic., Comment: Corrected the omission of the two series of exceptions involving the Ree groups; added a remark on the maximality of the intermediate subgroups in the exceptional cases

We impose a rather unknown algebraic structure called a ‘hyperstructure’ to the underlying space of an affine algebraic group scheme. This algebraic structure generalizes the classical group structure and is canonically defined by the structure of a Hopf algebra of global sections. This paper partially generalizes the result of A. Connes and C. Consani in [1] .

As a special type of factorization of finite groups, logarithmic signature (LS) is used as the main component of cryptographic keys for secret key cryptosystems such as PGM and public key cryptosystems like $$MST_1$$MST1, $$MST_2$$MST2 and $$MST_3$$MST3. An LS with the shortest length, called a minimal logarithmic signature (MLS), is even desirable for cryptographic applications. The MLS conjecture states that every finite simple group has an MLS. Recently, the conjecture has been shown to be true for general linear groups $$GL_n(q)$$GLn(q), special linear groups $$SL_n(q)$$SLn(q), and symplectic groups $$Sp_n(q)$$Spn(q) with q a power of primes and for orthogonal groups $$O_n(q)$$On(q) with q a power of 2. In this paper, we present new constructions of minimal logarithmic signatures for the orthogonal group $$O_n(q)$$On(q) and $$SO_n(q)$$SOn(q) with q a power of an odd prime. Furthermore, we give constructions of MLSs for a type of classical groups--the projective commutator subgroup $$P{\varOmega }_n(q)$$PΩn(q).

In this paper, we investigate the local Gan–Gross–Prasad conjecture for some pair of representations of U ( 3 ) × U ( 2 ) involving a non-generic representation. For a pair of generic L-parameters of ( U ( n ) , U ( n − 1 ) ) , it is known that there is a unique pair of representations in their associated Vogan L-packets which produces the unique Bessel model of these L-parameters. We showed that this is not true for some pair of L-parameters involving a non-generic one. On the other hand, we give the precise local theta correspondence for ( U ( 1 ) , U ( 3 ) ) not at the level of L-parameters but of individual representations in the framework of the local Langlands correspondence for unitary group. As an application of these results, we prove an analog of Ichino–Ikeda conjecture for some non-tempered case. The main tools in this work are the see-saw identity, local theta correspondence for (almost) equal rank cases and recent results on the local Gan–Gross–Prasad conjecture both on the Fourier–Jacobi and the Bessel case.

We classify Alt n and its perfect covers for n ≥ 6 up to conjugacy as subgroups of the exceptional complex Lie groups (with some exceptions where we give lower bounds and discuss the possibilities of Lie primitive subgroups).

We begin to generalize to good prime characteristic the results of Kraft and Procesi in their 1982 paper, “On the geometry of conjugacy classes in classical groups,” that describe which nilpotent orbits of an orthogonal or symplectic Lie algebra have normal closure. The first step is to modify the proof of a dimension formula for orthosymplectic orbits to be independent of the characteristic of the base field, excluding characteristic 2. This is accomplished by explicit construction of an appropriate cocharacter, which we show exists in the particular setting of the problem, relying heavily on the description of orthosymplectic orbits via ab-diagrams, an analogue of Young diagrams.

We continue study of subgroups of a Chevalley group $G_P(\Phi,R)$ over a ring $R$ with a root system $\Phi$ and a weight lattice $P$, containing the elementary subgroup $E_P(\Phi,K)$ over a subring $K$ of $R$. Recently A. Bak and A. Stepanov considered the symplectic case (i. e. the case of simply connected group of type $\Phi=C_l$) in characteristic 2. In this article we extend their result for groups with arbitrary weight lattice of types $B_l$ and $C_l$. Similarly to the work of Nuzhin that handles the case of an algebraic extension $R$ of a nonperfect field $K$ of bad characteristic, we use in the description a special kind of carpet subgroups. In the second half of the article we study Bruhat and Gauss decompositions for these carpet subgroups., Comment: The 1st version was in Russian, the current version is in English with minor corrections

We establish a Siegel-Weil formula for classical groups over a function field with odd characteristic, which asserts in many cases that the Siegel Eisenstein series is equal to an integral of a theta function. This is a function-field analogue of the classical result proved by A. Weil in his 1965 Acta Math. paper. We also give a convergence criterion for the theta integral by using Harder's reduction theory over function fields., Comment: some corrections and modifications are made in this version

Let $W$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$ of any characteristic and $mW$ denote the direct sum of $m$ copies of $W$. Let $\mathbb{F}_q[mW]^{{\rm GL}(W)}$ and $\mathbb{F}_q(mW)^{{\rm GL}(W)}$ denote the vector invariant ring and vector invariant field respectively where ${\rm GL}(W)$ acts on $W$ in the standard way and acts on $mW$ diagonally. We prove that there exists a set of homogeneous invariant polynomials $\{f_{1},f_{2},\ldots,f_{mn}\}\subseteq \mathbb{F}_q[mW]^{{\rm GL}(W)}$ such that $\mathbb{F}_q(mW)^{{\rm GL}(W)}=\mathbb{F}_q(f_{1},f_{2},\ldots,f_{mn}).$ We also prove the same assertions for the special linear groups and the symplectic groups in any characteristic, and the unitary groups and the orthogonal groups in odd characteristic., Comment: 15 pages; some errors have been corrected

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite almost simple primitive permutation group, with socle $G_0$ and point stabilizer $H$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$, denoted $b(G)$, is the minimal size of a base. We say that $G$ is standard if $G_0 = A_n$ and $\Omega$ is an orbit of subsets or partitions of $\{1, \ldots, n\}$, or if $G_0$ is a classical group and $\Omega$ is an orbit of subspaces (or pairs of subspaces) of the natural module for $G_0$. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have $b(G) \leqslant 7$ for every non-standard group $G$, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$ in its natural action on $24$ points. In this paper, we extend this result by classifying the non-standard groups with $b(G)=6$. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type., Comment: 27 pages; to appear in J. Algebra

We show that the minimal base size b ( G ) of a finite primitive permutation group G of degree n is at most 2 ( log ⁡ | G | / log ⁡ n ) + 24 . This bound is asymptotically best possible since there exists a sequence of primitive permutation groups G of degrees n such that b ( G ) = ⌊ 2 ( log ⁡ | G | / log ⁡ n ) ⌉ − 2 and b ( G ) is unbounded. As a corollary we show that a primitive permutation group of degree n that does not contain the alternating group Alt ( n ) has a base of size at most max ⁡ { n , 25 } .

In this paper we introduce two general identities relating Eisenstein series on split classical groups, as well as double covers of symplectic groups. The first identity can be viewed as an extension of the doubling construction introduced in [CFGK19] . The second identity is a generalization of the descent construction studied in [GRS11] .

In 1993, Wan obtained Anzahl theorems of subspaces with type in geometry of singular classical groups. As a generalization, we introduce the concept of geometry of t-singular classical groups, and make some new Anzahl theorems of subspaces with type. As applications, we determine the suborbits of subspaces under t-singular classical groups, and compute the lengths and ranks of these suborbits; show that the isotropic subspace poset of has normalized matching property and q-weighted log-concavity ordered by inclusion; and prove that the set of subspaces which are intersection of subspaces in an orbit of totally isotropic subspaces is a partition strongly regularized semilattice ordered by inclusion, and therefore obtain an analog of EKR theorem in this semilattice.

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally compact group which has a compact quotient group modulo its identity component and thus, in particular, each compact and each connected locally compact group; it also includes all locally compact abelian groups. This paper provides an overview of the structure theory and Lie theory of pro-Lie groups including results more recent than those in the authors' reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function which links the two. (A topological vector space is weakly complete if it is isomorphic to a power $\R^X$ of an arbitrary set of copies of $\R$. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected with pro-Lie groups., Comment: 19 pages

We use statistical mechanics--variants of the six-vertex model in the plane studied by means of the Yang---Baxter equation--to give new deformations of Weyl's character formula for classical groups of Cartan type B, C, and D, and a character formula of Proctor for type BC. In each case, the corresponding Boltzmann weights are associated with the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types A and C by Tokuyama and Hamel-King, respectively. A special case for classical types recovers deformations of the Weyl denominator formula due to Okada.

If $\mathscr{J}$ is a finite-dimensional nilpotent algebra over a finite field $\Bbbk$, the algebra group $P = 1+\mathscr{J}$ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If $\mathscr{J}$ is endowed with an involution $\widehat{\varsigma}$, then $\widehat{\varsigma}$ naturally defines a group automorphism of $P = 1+\mathscr{J}$, and we may consider the fixed point subgroup $C_{P}(\widehat{\varsigma}) = \{x\in P : \widehat{\varsigma}(x) = x^{-1}\}$. Assuming that $\Bbbk$ has odd characteristic $p$, we use the standard supercharacter theory for $P$ to construct a supercharacter theory for $C_{P}(\widehat{\varsigma})$. In particular, we obtain a supercharacter theory for the Sylow $p$-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by Andr\'e and Neto for the special case of the symplectic and orthogonal groups., Comment: Accepted for publication in the Journal of Algebra. arXiv admin note: text overlap with arXiv:1201.1060 by other authors