Kazhdan–Lusztig polynomials of boolean elements

We give closed combinatorial product formulas for Kazhdan–Lusztig polynomials and their parabolic analogue of type q in the case of boolean elements, introduced in (Marietti in J. Algebra 295:1–26, 2006), in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements.     

Leading coefficients of Kazhdan–Lusztig polynomials for Deodhar elements

We show that the leading coefficient of the Kazhdan–Lusztig polynomial P _x,w (q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in Billey and Warrington (J. Algebraic Combin. 13(2):111–136, [2001]) and Billey and Jones (Ann. Comb. [2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar’s algorithm (Deodhar in Geom. Dedicata 63(1):95–119, [1990]), we provide some combinatorial criteria to determine when μ(x,w)=1 for such permutations w. The author received support from NSF grants DMS-9983797 and DMS-0636297.     

Leading coefficients of Kazhdan–Lusztig polynomials and fully commutative elements

Let W be a Coxeter group of type . We show that the leading coefficient, μ(x,w), of the Kazhdan–Lusztig polynomial P _x,w is always equal to 0 or 1 if x is fully commutative (and w is arbitrary).     

Characterization of the classical domains by Caratheodory volume elements and Eisenman-Kobayashi volume elements

We give a sufficient and necessary condition that a domain is biholomorphic to the classical domain.     

Characterization of the classical domains by Carathèodory volume elements and Eisenman-Kobayashi volume elements

We give a sufficient and necessary condition that a domain is biholomorphic to the classical domain.     

The genus polynomials of cross-ladder digraphs in orientable surfaces

Some results about the genus distributions of graphs are known,but little is known about those of digraphs.In this paper,the method of joint trees initiated by Liu is generalized to compute the embedding genus distributions of digraphs in orientable surfaces.The genus polynomials for a new kind of 4-regular digraphs called the cross-ladders in orientable surfaces are obtained.These results are close to solving the third problem given by Bonnington et al.     

Membership of distributions of polynomials in the Nikolskii–Besov class

The main result of this paper asserts that the distribution density of any non-constant polynomial f(ξ₁,ξ₂,...) of degree d in independent standard Gaussian random variables ξ₁ (possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space B^1/d (R¹) of fractional order 1/d (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of k polynomials of degree d, also with a fractional order that is independent of the number of variables, but depends only on the degree d and the number of polynomials. We also give a new simple sufficient condition for a measure on R^k to possess a density in the Nikol’skii–Besov class B^α(R)^k. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on R^k via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.     

Pieri formulas for Macdonald��s spherical functions and polynomials

We present explicit Pieri formulas for Macdonald??s spherical functions (or generalized Hall-Littlewood polynomials associated with root systems) and their q-deformation the Macdonald polynomials. For the root systems of type A, our Pieri formulas recover the well-known Pieri formulas for the Hall-Littlewood and Macdonald symmetric functions due to Morris and Macdonald as special cases.     

Higher asymptotics of unitarity in ��quantization commutes with reduction��

Let M be a compact K?hler manifold equipped with a Hamiltonian action of a compact Lie group G. Guillemin and Sternberg (Invent Math 67:515?C538, 1982, no. 3), showed that there is a geometrically natural isomorphism between the G-invariant quantum Hilbert space over M and the quantum Hilbert space over the symplectic quotient M //G. This map, though, is not in general unitary, even to leading order in ${\hslash}$ . Hall and Kirwin (Commun Math Phys 275:401?C422, 2007, no. 2), showed that when the metaplectic correction is included, one does obtain a map which, while not in general unitary for any fixed ${\hslash}$ , becomes unitary in the semiclassical limit ${\hslash\rightarrow0}$ (cf. the work of Ma and Zhang (C R Math Acad Sci Paris 341:297?C302, 2005, no. 5), and (Astérisque No. 318:viii+154, 2008). The unitarity of the classical Guillemin?CSternberg map and the metaplectically corrected analogue is measured by certain functions on the symplectic quotient M //G. In this paper, we give precise expressions for these functions, and compute complete asymptotic expansions for them as ${\hslash\rightarrow0}$ .     

Approximation of analytic functions by polynomials “close” to polynomials of best approximation

The rate of approximation of analytic functions at interior points of compact sets with connected complement by polynomials close to polynomials of best approximation is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 208–214, February, 1992.     

When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Given {P_n}_n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e., 

     

Near-rings of polynomials overΩ-groups

IfV is a variety of -groups and ifGV then one can study the algebraG ^V [x] of polynomials overG andx inV. With respect to addition and substitution,G ^V [x] is a near-ring. Its zero-symmetric part can excellently be used to describe generated ideals. Also, we study maximal left ideals and get a general result on the structure ofG ^V [V].This study was kindly supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (Projekt Nr. 3479).