Search

In this paper we study algebro-geometric aspects of the Christoffel-Darboux kernels for classical orthogonal polynomials with rational coefficients. We find a novel connection between a projective curve defined by theChristoffel-Darboux kernel and a system of Diophantine equations, which was originally designed by Hausdorff (1909) for applications to Waring’s problem, and which is closely related to quadrature formulas in numerical analysis and Gaussian designs in algebraic combinatorics. We prove some nonsolvability results of such Hausdorff-type equations.

In this paper, we give a representation-theoretic interpretation of the specialization E w ∘ λ ( q , ∞ ) E_{w_\circ \lambda } (q,\infty ) of the nonsymmetric Macdonald polynomial E w ∘ λ ( q , t ) E_{w_\circ \lambda }(q,t) at t = ∞ t=\infty in terms of the Demazure submodule V w ∘ − ( λ ) V_{w_\circ }^- (\lambda ) of the level-zero extremal weight module V ( λ ) V(\lambda ) over a quantum affine algebra of an arbitrary untwisted type. Here, λ \lambda is a dominant integral weight, and w ∘ w_\circ denotes the longest element in the finite Weyl group W W . Also, for each x ∈ W x \in W , we obtain a combinatorial formula for the specialization E x λ ( q , ∞ ) E_{x \lambda } (q, \infty ) at t = ∞ t=\infty of the nonsymmetric Macdonald polynomial E x λ ( q , t ) E_{x \lambda } (q,t) and also a combinatorical formula for the graded character gch ⁡ V x − ( λ ) \operatorname {gch} V_{x}^- (\lambda ) of the Demazure submodule V x − ( λ ) V_{x}^- (\lambda ) of V ( λ ) V(\lambda ) . Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape λ \lambda .

Let F n F_n be a free group of rank n n and F n N F_n^N the quotient group of F n F_n by a subgroup [ Γ n ( 3 ) , Γ n ( 3 ) ] [ [ Γ n ( 2 ) , Γ n ( 2 ) ] , Γ n ( 2 ) ] [\Gamma _n(3), \Gamma _n(3)][[\Gamma _n(2),\Gamma _n(2)],\Gamma _n(2)] , where Γ n ( k ) \Gamma _n(k) denotes the k k -th subgroup of the lower central series of the free group F n F_n . In this paper, we determine the group structure of the graded quotients of the lower central series of the group F n N F_n^N by using a generalized Chen’s integration in free groups. Then we apply it to the study of the Johnson homomorphisms of the automorphism group of F n F_n . In particular, under taking a reduction of the target of the Johnson homomorphism induced from a quotient map F n → F n N F_n \rightarrow F_n^N , we see that there appear only two irreducible components, the Morita obstruction S k H Q S^k H_{\mathbf {Q}} and the Schur-Weyl module of type H Q [ k − 2 , 1 2 ] H_{\mathbf {Q}}^{[k-2, 1^2]} , in the cokernel of the rational Johnson homomorphism τ k , Q ′ = τ k ′ ⊗ i d Q \tau _{k, \mathbf {Q}}’=\tau _k’ \otimes \mathrm {id}_{\mathbf {Q}} for k ≥ 5 k \geq 5 and n ≥ k + 2 n \geq k+2 .

In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type. This conjecture was proved for closed S in [WZ2,WZ3,FH,M4] and for open S in [M4]. It was proved for the other orbits in [M5] when G_R is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed K_C-orbit when G_R is of Hermitian type. Thus the conjecture is completely solved affirmatively., 15 pages

We present a representation theoretic description of the non-empty strata in the Hesselink stratification of the nullcone of representations of quivers. We use this stratification to define optimal filtrations on representations of finite dimensional algebras. As an application we investigate the isomorphism problem for uniserial representations. 1. OPTIMAL FILTRATIONS 1.1. Let A be a finite dimensional algebra over an algebraically closed field k. In this paper we want to parameterize isomorphism classes of finite dimensional Amodules having a specific Jordan-Holder sequence. In particular, we want to relate the recent results of K. Bongartz and B. Huisgen-Zimmermann [3, 4, 1] on uniserial modules to the Hesselink stratification of nullcones. By Morita theory we may reduce to the case that A is a basic algebra. That is, all simple A-modules are one dimensional. In this case we can write A as the quotient of the path algebra of a quiver and relate finite dimensional A-modules to representations of this quiver. 1.2. A quiver Q is a directed graph on a finite set of vertices {vl,..., Vn}. Let aij be the number of directed arrows from vi to vj (or loops if v = vj). The Euler-form of Q is the bilinear form X: Z x Zn Z determined by the matrix X = (Xij)i,j c Mn(Z) with entries Xij = 6ij aij Clearly, X encodes the structure of the directed graph Q. A representation V of a quiver Q of dimension vector a = (al,..., an) E Nn assigns to every arrow vi -vj in Q a matrix V(Q) E Majixi (k). The set of all a-dimensional representations form an affine space

In this paper we give the étale local classification of Schelter-Procesi smooth orders in central simple algebras. In particular, we prove that if Δ \Delta is a central simple K K -algebra of dimension n 2 n^2 , where K K is a field of trancendence degree d d , then there are only finitely many étale local classes of smooth orders in Δ \Delta . This result is a non-commutative generalization of the fact that a smooth variety is analytically a manifold, and so has only one type of étale local behaviour.