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Motivated by statistical applications, this paper introduces Cauchy identities for characters of the compact classical groups. These identities generalize the well-known Cauchy identity for characters of the unitary group, which are Schur functions of symmetric function theory. Application to statistical hypothesis testing is briefly sketched.

The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of basic invariants for the classical groups GL$(V)$, O$(n)$, and Sp$(n)$ for $n$ even. In the first half of the paper we set up relevant definitions and theorems for our search for the set of basic invariants, starting with linear algebraic groups and then discussing associative algebras. We then state and prove a monumental theorem that will allow us to proceed with hope: it says that the set of basic invariants is finite if $G$ is reductive. Finally we state without proof the First Fundamental Theorems, which aim to list explicitly the relevant sets of basic invariants, for the classical groups above. We end by commenting on some applications of invariant theory, on the history of its development, and stating a useful theorem in the appendix whose proof lies beyond the scope of this work.

Let G denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for G: given dominant weights \lambda^1, ..., \lambda^r such that the tensor product V_{N\lambda^1} \otimes ... \otimes V_{N\lambda^r} contains nonzero G-invariants for some N \ge 1, we show that the tensor product V_{2\lambda^1} \otimes ... \otimes V_{2\lambda^r} also contains nonzero G-invariants. This extends results of Kapovich-Millson and Belkale-Kumar and complements similar results for the general linear group due to Knutson-Tao and Derksen-Weyman. Our techniques involve the invariant theory of quivers equipped with an involution and the generic representation theory of certain quivers with relations.

This paper presents a powerfull method to integrate general monomials on the classical groups with respect to their invariant (Haar) measure. The method has first been applied to the orthogonal group in [J. Math. Phys. 43, 3342 (2002)], and is here used to obtain similar integration formulas for the unitary and the unitary symplectic group. The integration formulas turn out to be of similar form. They are all recursive, where the recursion parameter is the number of column (row) vectors from which the elements in the monomial are taken. This is an important difference to other integration methods. The integration formulas are easily implemented in a computer algebra environment, which allows to obtain analytical expressions very efficiently. Those expressions contain the matrix dimension as a free parameter.

Let G be a unitary, symplectic or special orthogonal group over a locally compact non-archimedean local field of odd residual characteristic. We construct many new supercuspidal representations of G, and Bushnell-Kutzko types for these representations. Moreover, we prove that every irreducible supercuspidal representation of G arises from our constructions.

A holomorphic discrete series can be realized as the space of the holomorphic sections of a homogeneous vector bundle on a bounded symmetric domain. We can embed it into a degenerate principal series realized as the space of hyperfunction sections of a vector bundle on the Shilov boundary by taking the boundary value. We consider an algebraic version of higher cohomological analogue of such an embedding for complex reductive groups and classical groups of type A.

A holomorphic discrete series can be realized as the space of the holomorphic sections of a homogeneous vector bundle on a bounded symmetric domain. We can embed it into a degenerate principal series realized as the space of hyperfunction sections of a vector bundle on the Shilov boundary by taking the boundary value. We consider an algebraic version of higher cohomological analogue of such an embedding for complex reductive groups and classical groups of type A.

We consider several questions about restriction of representations of classical and metaplectic groups over local and global fields to subgroups, extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. This includes Bessel and Fourier-Jacobi models too. We formulate several conjectures about these restriction problems involving root numbers of symplectic representations in the local case, and central critical L-value in the global case. Along the way we prove several results both in number theory and representation theory.

This paper contains a complete description of classes of the unitary equivalence of the admissible representations of infinite-dimensional classic matrix groups paper.

In this paper we determine the classical simple groups of dimension r=3,5 which are (2,3)-generated (the cases r = 2, 4 are known). If r = 3, they are PSL_3(q), q 4, and PSU_3(q^2), q^2 9, 25. If r = 5 they are PSL_5(q), for all q, and PSU_5(q^2), q^2 >= 9. Also, the soluble group PSU_3(4) is not (2,3)-generated. We give explicit (2,3)-generators of the linear preimages, in the special linear groups, of the (2,3)-generated simple groups.

This paper defines and develops cycle indices for the finite classical groups. These tools are then applied to study properties of a random matrix chosen uniformly from one of these groups. Properties studied by this technique will include semisimplicity, regularity, regular semisimplicity, the characteristic polynomial, number of Jordan blocks, and average order of a matrix.

Algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real or quaternionic structure, it is natural to ask for the properties of the groups of real or quaternionic algebraic cycles on P(V). In this paper and its sequel the homotopy structure of these cycle groups is determined. They bear a direct relationship to characteristic classes for the classical groups, and functors in K-theory extend directly to these groups. These groups give rise to E-infinity-ring spaces, and the maps extending the K-theory functors are ring maps. The stabilized space of cycles is a product of (Z/2Z)-equivariant Eilenberg-MacLane spaces indexed by the representations R^{n,n} for n > 0. This gives a wide generalization of the results in Boyer, Lawson, Lima-Filho, Mann and Michelsohn on the Segal question. The ring structure on the homotopy groups of these stabilized spaces is explicitly computed. In the real case it is a quotient of a polynomial algebra on two generators corresponding to the first Pontrjagin and first Stiefel-Whitney classes. This yields an interesting total characteristic class for real bundles. It is a mixture of integral and mod 2 classes and has nice multiplicative properties. The class is shown to be the (Z/2Z)-equivariant Chern class on Atiyah's KR-theory.

Integrable systems underlying the Seiberg-Witten solutions for the N=2 SQCD with gauge groups SO(n) and Sp(n) are proposed. They are described by the inhomogeneous XXX spin chain with specific boundary conditions given by reflection matrices. We attribute reflection matrices to orientifold planes in the brane construction and briefly discuss its possible deformations.

The behavior of the images of a fixed element of order p in irreducible representations of a classical algebraic group in odd characteristic p with highest weights large enough with respect to p and this element is investigated. Lower estimates for the number of Jordan blocks of size p in images of such elements that lie in naturally embedded subgroups of the same type as the initial group and smaller ranks are obtained.

In this paper we study the (2,k)-generation of the finite classical groups SL(4,q), Sp(4,q), SU(4,q^2) and their projective images. Here k is the order of an arbitrary element of SL(2,q), subject to the necessary condition k>= 3. When q is even we allow also k=4.

We prove that every quasisimple group of classical type is a product of boundedly many conjugates of a quasisimple subgroup of type A_n.

We study components of the Bernstein category for a p-adic classical group (with p odd) with inertial support a self-dual positive level supercuspidal representation of a Siegel Levi subgroup. More precisely, we use the method of covers to construct a Bushnell-Kutzko type for such a component. A detailed knowledge of the Hecke algebra of the type should have number-theoretic implications.

We suggest a possible programme to associate geometric "flag-like" data to an arbitrary simple quantum group, in the spirit of the noncommutative algebraic geometry developed by Artin, Tate, and Van den Bergh. We then carry out this programme for the standard quantum SL(n) of Drinfel'd and Jimbo, where the varieties involved are certain T-stable subvarieties of the (ordinary) flag variety.

It has been suggested that quantum coherence in the selectivity filter of ion channel may play a key role in fast conduction and selectivity of ions. However, it has not been clearly elucidated yet why classical coherence is not sufficient for this purpose. In this paper, we investigate the classical vibrational coherence between carbonyl groups oscillations in the selectivity filter of KcsA ion channels based on the data obtained from molecular dynamics simulations. Our results show that classical coherence plays no effective role in fast ionic conduction.

In this paper we prove the Aubert-Baum-Plymen-Solleveld conjecture for the split classical groups and establish the connection with the Langlands correspondence. To do this, we review the notion of cuspidality for enhanced Langlands parameters and also review the notion of cuspidal support for enhanced Langlands parameters for split classical groups, both introduced by the author in earlier work.

In this paper we prove the Aubert-Baum-Plymen-Solleveld conjecture for the split classical groups and establish the connection with the Langlands correspondence. To do this, we review the notion of cuspidality for enhanced Langlands parameters and also review the notion of cuspidal support for enhanced Langlands parameters for split classical groups, both introduced by the author in earlier work.

In this paper we prove the Aubert-Baum-Plymen-Solleveld conjecture for the split classical groups and establish the connection with the Langlands correspondence. To do this, we review the notion of cuspidality for enhanced Langlands parameters and also review the notion of cuspidal support for enhanced Langlands parameters for split classical groups, both introduced by the author in earlier work.

In this paper we prove the Aubert-Baum-Plymen-Solleveld conjecture for the split classical groups and establish the connection with the Langlands correspondence. To do this, we review the notion of cuspidality for enhanced Langlands parameters and also review the notion of cuspidal support for enhanced Langlands parameters for split classical groups, both introduced by the author in earlier work.

We study algebras encoding stable range branching rules for the pairs of complex classical groups of the same type in the context of toric degenerations of spherical varieties. By lifting affine semigroup algebras constructed from combinatorial data of branching multiplicities, we obtain algebras having highest weight vectors in multiplicity spaces as their standard monomial type bases. In particular, we identify a family of distributive lattices and their associated Hibi algebras which can uniformly describe the stable range branching algebras for all the pairs we consider.

In constructive recognition of a representation of a Classical group $G$, much attention has been paid to the natural representation as well as to generic (Black Box) algorithms that treat all representations uniformly. There are theoretical and practical improvements to be made by giving special treatment to certain non-natural representations that arise frequently. In this paper we present and analyse a Las Vegas algorithm for rewriting the Symmetric Square representation.

Let $C(T)$ be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either $B_n$ or $D_n$. Let $C_Y(T)$ be a natural quotient of $C(T)$, and if $C(T)$ is simply-laced (which means all the relations between the generators has order 2 or 3), $C_Y(T)$ is a generalized Coxeter group, too . Let $A_{t,n}$ be a group which contains $t$ Abelian groups generated by $n$ elements. The main result in this paper is that $C_Y(T)$ is isomorphic to $A_{t,n} \semidirect B_n$ or $A_{t,n} \semidirect D_n$, depends on whether the signed graph $T$ contains loops or not, or in other words C(T) is simply-laced or not, and $t$ is the number of the cycles in $T$. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups.

The aim of this work is using the information in the ATLAS of Finite Groups (Wilson et al., 1985) and by developing a program inside the GAP computational system (The GAP computational System , 2010), to determine all Hall π -subgroups for some finite classical simple groups such as some of the finite unitary and finite simplectic simple groups and some of finite simple groups of of Lie type. The structures and permutation representations of the Hall π -subgroups have been found. By using the following theoretical and computational algorithms, we determined the solvable subgroups of large order of the finite non-abelian simple linear groups G = L2(p) = PSL (2, p), for p ≥ 5 and p is a prime number, also their presentations and permutation representations have been found.

We consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical groups instead of a product of the general linear groups and by considering the dual action of groups on "vertex" vector spaces together with the usual action. A generating system for the corresponding algebra of invariants is found. In particular, a generating system for the algebra of SO(n)-invariants of several matrices is constructed over a field of characteristic different from 2. The proof uses the reduction to semi-invariants of mixed representations of a quiver and the decomposition formula that generalizes Amitsur's formula for the determinant.

We determine the ranks of the Sylow 2-subgroups of the classical simple groups of odd characteristic.

We determine the ranks of the Sylow 2-subgroups of the classical simple groups of odd characteristic.

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion which involves root systems.

In part I of this work we studied the spaces of real algebraic cycles on a complex projective space P(V), where V carries a real structure, and completely determined their homotopy type. We also extended some functors in K-theory to algebraic cycles, establishing a direct relationship to characteristic classes for the classical groups, specially Stiefel-Whitney classes. In this sequel, we establish corresponding results in the case where V has a quaternionic structure. The determination of the homotopy type of quaternionic algebraic cycles is more involved than in the real case, but has a similarly simple description. The stabilized space of quaternionic algebraic cycles admits a nontrivial infinite loop space structure yielding, in particular, a delooping of the total Pontrjagin class map. This stabilized space is directly related to an extended notion of quaternionic spaces and bundles (KH-theory), in analogy with Atiyah's real spaces and KR-theory, and the characteristic classes that we introduce for these objects are nontrivial. The paper ends with various examples and applications.

We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of any dimension invariant and apply these transformations to field equations.

We determine the vacuum structure of N=2 supersymmetric QCD with fundamental quarks for gauge groups SO(n) and Sp(2n), extending prior results for SU(n). The solutions are all given in terms of families of hyperelliptic Riemann surfaces of genus equal to the rank of the gauge group. In the scale invariant cases, the solutions all have exact S-dualities which act on the couplings by subgroups of PSL(2,Z) and on the masses by outer automorphisms of the flavor symmetry. They are shown to reproduce the complete pattern of symmetry breaking on the Coulomb branch and predict the correct weak--coupling monodromies. Simple breakings with squark vevs provide further consistency checks involving strong--coupling physics.

This paper provides new, relatively simple proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive, for good characteristic, the description of the natural partial order of unipotent classes of a classical group in terms of their Jordan blocks. This result was originally due to Gerstenhaber and Hesselink. As a consequence we obtain a proof of the Bala--Carter Theorem which holds even in certain bad characteristics (this proof requires the prior classification of unipotent classes, unlike the original proofs due to Bala, Carter and Pommerening).

We give a constructive account of the fundamental ingredients of Poisson Lie theory as the basis for a description of the classical double group $D$. The double of a group $G$ has a pointwise decomposition $D\sim G\times G^*$, where $G$ and $G^*$ are Lie subgroups generated by dual Lie algebras which form a Lie bialgebra. The double is an example of a factorisable Poisson Lie group, in the sense of Reshetikhin and Semenov-Tian-Shansky [1], and usually the study of its Poisson structures is developed only in the case when the subgroup $G$ is itself factorisable. We give an explicit description of the Poisson Lie structure of the double without invoking this assumption. This is achieved by a direct calculation, in infinitesimal form, of the dressing actions of the subgroups on each other, and provides a new and general derivation of the Poisson Lie structure on the group $G^*$. For the example of the double of SU(2), the symplectic leaves of the Poisson Lie structures on SU(2) and SU(2$)^*$ are displayed.

We give a constructive account of the fundamental ingredients of Poisson Lie theory as the basis for a description of the classical double group $D$. The double of a group $G$ has a pointwise decomposition $D\sim G\times G^*$, where $G$ and $G^*$ are Lie subgroups generated by dual Lie algebras which form a Lie bialgebra. The double is an example of a factorisable Poisson Lie group, in the sense of Reshetikhin and Semenov-Tian-Shansky [1], and usually the study of its Poisson structures is developed only in the case when the subgroup $G$ is itself factorisable. We give an explicit description of the Poisson Lie structure of the double without invoking this assumption. This is achieved by a direct calculation, in infinitesimal form, of the dressing actions of the subgroups on each other, and provides a new and general derivation of the Poisson Lie structure on the group $G^*$. For the example of the double of SU(2), the symplectic leaves of the Poisson Lie structures on SU(2) and SU(2$)^*$ are displayed.

We give a constructive account of the fundamental ingredients of Poisson Lie theory as the basis for a description of the classical double group $D$. The double of a group $G$ has a pointwise decomposition $D\sim G\times G^*$, where $G$ and $G^*$ are Lie subgroups generated by dual Lie algebras which form a Lie bialgebra. The double is an example of a factorisable Poisson Lie group, in the sense of Reshetikhin and Semenov-Tian-Shansky [1], and usually the study of its Poisson structures is developed only in the case when the subgroup $G$ is itself factorisable. We give an explicit description of the Poisson Lie structure of the double without invoking this assumption. This is achieved by a direct calculation, in infinitesimal form, of the dressing actions of the subgroups on each other, and provides a new and general derivation of the Poisson Lie structure on the group $G^*$. For the example of the double of SU(2), the symplectic leaves of the Poisson Lie structures on SU(2) and SU(2$)^*$ are displayed.

In this paper we introduce a new method for manufacturing harmonic morphisms from semi-Riemannian manifolds. This is employed to yield a variety of new examples from the compact Lie groups SO(n), SU(n) and Sp(n) equipped with their standard Riemannian metrics. We develop a duality principle and show how this can be used to construct the first known examples of harmonic morphisms from the non-compact Lie groups SL(n,R), SU(2n), Sp(n,R), SO(2n), SO(p,q), SU(p,q) and Sp(p,q) equipped with their standard dual semi-Riemannian metrics.

The main aim of this work is to construct several new families of proper biharmonic functions defined on open subsets of the classical compact simple Lie groups $\SU n$, $\SO n$ and $\Sp n$. We work in a geometric setting which connects our study with the theory of submersive harmonic morphisms. We develop a general duality principle and use this to interpret our new examples on the Euclidean sphere $\s ^3$ and on the hyperbolic space $\H^3$.

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively on a set of size n > 1, then the proportion of derangements in G is greater than delta. We show that with possibly finitely many exceptions, one can take delta = .016. Indeed, we prove much stronger results showing that for many actions, the proportion of derangements goes to 1 as n increases and prove similar results for families of permutation representations.

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively on a set of size n > 1, then the proportion of derangements in G is greater than delta. We show that with possibly finitely many exceptions, one can take delta = .016. Indeed, we prove much stronger results showing that for many actions, the proportion of derangements goes to 1 as n increases and prove similar results for families of permutation representations.

The character ring \CGL of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra \Sym$ of symmetric functions. Here we study the character rings \CO and \CSp of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that \CO and \CSp also admit natural Hopf algebra structures that are isomorphic to that of \CGL, and hence to \Sym. The isomorphisms are determined explicitly, along with the specification of standard bases for \CO and \CSp analogous to those used for \Sym. A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur-Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the \CGL case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras \CO and \CSp are not self-dual. The dual Hopf algebras \CO^* and \CSp^* are identified. Finally, the Hopf algebra of the universal rational character ring \CGLrat of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.

We continue our study of the poles of local Langlands L-functions through the theory of induced from supercuspidal representations of quasi-split groups. Here we study the odd special orthogonal groups, and hence determine poles of Rankin product L-functions. The pole of the intertwining operator is determined in terms of the theory of orbital integrals. This gives a description of the poles in terms of twisted endoscopy, as in previous cases. We use the language of functorial transfer to give precise descrption of the pole in terms of the local components of the global transfer, which has now been established.

We study the middle convolution of local systems on the punctured affine line in the setting of singular cohomology and in the setting of \'etale cohomology. We derive a formula to compute the topological monodromy of the middle convolution in the general case and use it to deduce some irreducibility criteria. Then we give a geometric interpretation of the middle convolution in the \'etale setting. This geometric approach to the convolution and the theory of Hecke characters yields information on the occurring arithmetic determinants. We employ these methods to realize special linear groups regularly as Galois groups over ${\bf Q}(t).$

We compute the irreducible constitutents of the product of the Weil character and the Steinberg character in those finite classical groups for which a Weil character is defined, namely the symplectic, unitary and general linear groups. It turns out that this product is multiplicity free for the symplectic and general unitary groups, but not for the general linear groups. As an application we show that the restriction of the Steinberg character of such a group to the subgroup stabilizing a vector in the natural module is multiplicity free. The proof of this result for the unitary groups uses an observation of Brunat, published as an appendix to our paper. As our "Weil character" for the symplectic groups in even characteristic we use the 2-modular Brauer character of the generalized spinor representation. Its product with the Steinberg character is the Brauer character of a projective module. We also determine its indecomposable direct summands.

Motivated by analogous results for the symmetric group and compact Lie groups, we study the distribution of the number of fixed vectors of a random element of a finite classical group. We determine the limiting moments of these distributions, and find exactly how large the rank of the group has to be in order for the moment to stabilize to its limiting value. The proofs require a subtle use of some q-series identities. We also point out connections with orthogonal polynomials.

In this paper we compute the homology of the braid groups, with coefficients in the module Z[q^+-1] given by the ring of Laurent polynomials with integer coefficients and where the action of the braid group is defined by mapping each generator of the standard presentation to multiplication by -q. The homology thus computed is isomorphic to the homology with constant coefficients of the Milnor fiber of the discriminantal singularity.

In this paper we compute the homology of the braid groups, with coefficients in the module Z[q^+-1] given by the ring of Laurent polynomials with integer coefficients and where the action of the braid group is defined by mapping each generator of the standard presentation to multiplication by -q. The homology thus computed is isomorphic to the homology with constant coefficients of the Milnor fiber of the discriminantal singularity.

Let $M$ be a random matrix in the orthogonal group $\O_n$, distributed according to Haar measure, and let $A$ be a fixed $n\times n$ matrix over $\R$ such that $\tr(AA^t)=n$. Then the total variation distance of the random variable $\tr(AM)$ to standard normal is bounded by $2\sqrt{3}/(n-1)$, and this rate is sharp up to the constant. Analogous results are obtained for $M$ a random unitary matrix and $A$ a fixed $n\times n$ matrix over $\C$. The proofs are applications of a new abstract normal approximation theorem which extends Stein's method of exchangeable pairs to situations in which continuous symmetries are present.