Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of \((\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})\) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials.     

A numerical study on exceptional eigenvalues of certain congruence subgroups of \mathrm {SO}(n,1) and \mathrm {SU}(n,1)

In a previous work, we applied lattice point theorems on hyperbolic spaces to obtain asymptotic formulas for the number of integral representations of negative integers by quadratic and Hermitian forms of signature \((n,1)\) lying in Euclidean balls of increasing radius. That formula involved an error term that depended on the first nonzero eigenvalue of the Laplace–Beltrami operator on the corresponding congruence hyperbolic manifolds. The aim of this paper is to compare the error term obtained by experimental computations with the error term mentioned above, for several choices of quadratic and Hermitian forms. Our numerical results provide evidence of the existence of exceptional eigenvalues for some arithmetic subgroups of \(\mathrm {SU}(3,1)\) , \(\mathrm {SU}(4,1)\) , and \(\mathrm {SU}(5,1)\) , and thus they contradict the generalized Selberg (and Ramanujan) conjecture in these cases. Furthermore, for several arithmetic subgroups of \(\mathrm {SO}(4,1)\) , \(\mathrm {SO}(6,1)\) , \(\mathrm {SO}(8,1)\) , and \(\mathrm {SU}(2,1)\) , there is evidence of a lower bound on the first nonzero eigenvalue that is better than the already known lower bound for congruences subgroups.     

Ramanujan-style proof of $$p_{-3}(11n+7) \equiv 0\ (\mathrm{mod\ }11)$$

In this note, we establish two identities of \((q;\,q)_\infty ^8\) by using Jacobi’s four-square theorem and two of Ramanujan’s identities. As an important consequence, we present one Ramanujan-style proof of the congruence \(p_{-3}(11n+7)\equiv 0\ (\mathrm{mod\ }11)\), where \(p_{-3}(n)\) denotes the number of 3-color partitions of n.     

Subconvexity for sup-norms of cusp forms on $$\mathrm{PGL}(n)$$

Let F be an \(L^2\)-normalized Hecke Maaß cusp form for \(\Gamma _0(N) \subseteq {\mathrm{SL}}_{n}({\mathbb {Z}})\) with Laplace eigenvalue \(\lambda _F\). If \(\Omega \) is a compact subset of \(\Gamma _0(N)\backslash {\mathrm{PGL}}_n/\mathrm{PO}_{n}\), we show the bound \(\Vert F|_{\Omega }\Vert _{\infty } \ll _{ \Omega } N^{\varepsilon } \lambda _F^{n(n-1)/8 - \delta }\) for some constant \(\delta = \delta _n> 0\) depending only on n.     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

Combinatorial invariants for nets of conics in $$\mathrm {PG}(2,q)$$ PG ( 2 , q )

Designs, Codes and Cryptography - The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics...     

A Clifford Algebraic Framework for $$\mathfrak{sp}(m)$$-Invariant Differential Operators

We introduce a framework for studying differential operators which are invariant with respect to the real (complex) symplectic Lie algebra ( ), associated to a quaternionic structure on a vector space . To do so, these algebras are realized within the orthogonal Lie algebra . This leads in a natural way to a refinement of the recently introduced notion of complex Hermitean Clifford analysis, in which four variations of the classical Dirac operator play a dominant role. David Eelbode: Postdoctoral fellow supported by the F.W.O. Vlaanderen (Belgium).     

Polynomials for \mathrm{GL}_p\times \mathrm{GL}_q orbit closures in the flag variety

The subgroup \(K=\mathrm{GL}_p \times \mathrm{GL}_q\) of \(\mathrm{GL}_{p+q}\) acts on the (complex) flag variety \(\mathrm{GL}_{p+q}/B\) with finitely many orbits. We introduce a family of polynomials specializing representatives for cohomology classes of the orbit closures in the Borel model. We define and study \(K\) -orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the \(H\) -polynomials and the Kazhdan–Lusztig–Vogan polynomials.     

Brauer algebras of type \mathrm{I}_2^n(n\ge 5)

We will present an algebra related to the Coxeter group of type \(\mathrm{I}_2^n\) which can be taken as the twisted subalgebra in Brauer algebra of type \(\mathrm{A}_{n-1}\) . Also, we will describe some properties of this algebra.     

\mathcal{T}(5) families of overlapping disks

We prove the following: If a finite family of unit (radius) disks has the property that the distance between every pair of centres is greater than 4/3 and every subset of at most five disks has a common transversal line, then all disks have a common transversal line.     

On the generalized $$text {SO}(2n,{{mathbb {C}}})$$ SO ( 2 n,C ) -opers

Annals of Global Analysis and Geometry - Since their introduction by Beilinson–Drinfeld (Opers, 1993. arXiv math/0501398; Quantization of Hitchin’s integrable system and Hecke...     

ExponentialL 2-convergence of quantum Markov semigroups on\mathcal{B}(h)

With a quantum Markov semigroup (Τ _t ) _t≥0 on , whichhas a faithful normal invariant state ρ, we associate semigroupsT ^(s) (s∈[0],[1]) on the set of Hilbert-Schmidt operators onh defined by the rule . This allows us to use spectral theory to study the infinitesimal generatorL ^(s) of the semigroupT ^(s) and deduce information on the rate of the decay to equilibrium of Τ by means of estimates of the spectral gap ofL ^(s) . Fors=1/2, this method is applied to a class of quantum Markov semigroups on . We prove simple but reasonably general sufficient conditions, as well as necessary and sufficient conditions, for the gap(L ^(1/2)) to be positive. The exact value of the gap(L ^(1/2)) is computed or estimated for a certain class of equations motivated by classical probability or physical applications. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 523–538, October, 2000.     

Toward logarithmic extensions of $$\widehat{s\ell }(2)_k $$ conformal field models

For positive integers p = k + 2, we construct a logarithmic extension of the conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a butterfly resolution of a three-boson realization of . The currents W⁻(z) and W⁺(z) of a W-algebra acting in the kernel are determined by a highest-weight state of dimension 4p − 2 and charge 2p − 1 and by a (θ=1)-twisted highest-weight state of the same dimension 4p − 2 and opposite charge −2p+1. We construct 2p W-algebra representations, evaluate their characters, and show that together with the p−1 integrable representation characters, they generate a modular group representation whose structure is described as a deformation of the (9p−3)-dimensional representation R _p+1⊕ℂ²⊗R _p+1ʕR _p−1⊕ℂ² R _p−1⊕ℂ³ R _p−1, where R _p−1 is the SL(2, ℤ)-representation on integrable-representation characters and R _p+1 is a (p+1)-dimensional SL(2, ℤ)-representation known from the logarithmic (p, 1) model. The dimension 9p − 3 is conjecturally the dimension of the space of torus amplitudes, and the ℂⁿ with n = 2 and 3 suggest the Jordan cell sizes in indecomposable W-algebra modules. We show that under Hamiltonian reduction, the W-algebra currents map into the currents of the triplet W-algebra of the logarithmic (p, 1) model. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 291–346, December, 2007.     

Twisted cubic and point-line incidence matrix in $$\mathrm {PG}(3,q)$$ PG ( 3 , q )

Designs, Codes and Cryptography - We consider the structure of the point-line incidence matrix of the projective space $$\mathrm {PG}(3,q)$$ connected with orbits of points and lines under the...     

Diagonal reduction algebra for $$\mathfrak{osp}(1|2)$$

Theoretical and Mathematical Physics - The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $$( \mathfrak{G} , \mathfrak{g} )$$ has previously...     

Parametrization of \text{ PSL }(2,\mathbb C )-representations of surface groups

For an oriented surface of genus $g$ with $b$ boundary components, we construct a rational map from a subset of $\mathbb{C }^{6g-6+3b}$ onto an open algebraic subset of the $\text{ PSL }(2,\mathbb C )$ -character variety as an analogue of the Fenchel-Nielsen coordinates. After taking the quotient by an action of a finite group, we obtain a parametrization of a subset of the $\text{ PSL }(2,\mathbb C )$ -character variety, and similarly for the $\text{ SL }(2,\mathbb C )$ -character variety. We can systematically calculate a set of matrix generators by rational functions of the parameters. We give transformation formulae under elementary moves of pants decompositions.