1,435 research outputs found
Inequalities on Bruhat graphs, R- and Kazhdan-Lusztig polynomials
From a combinatorial perspective, we establish three inequalities on
coefficients of - and Kazhdan-Lusztig polynomials for crystallographic
Coxeter groups: (1) Nonnegativity of -coefficients of -polynomials,
(2) a new criterion of rational singularities of Bruhat intervals by sum of
quadratic coefficients of -polynomials, (3) existence of a certain strict
inequality (coefficientwise) of Kazhdan-Lusztig polynomials. Our main idea is
to understand Deodhar's inequality in a connection with a sum of
-polynomials and edges of Bruhat graphs.Comment: 16 page
ENUMERATIVE COMBINATORICS ON DETERMINANTS AND SIGNED BIGRASSMANNIAN POLYNOMIALS
As an application of linear algebra for enumerative combinatorics,
we introduce two new ideas, signed bigrassmannian polynomials
and bigrassmannian determinant. First, a signed bigrassmannian
polynomial is a variant of the statistic given by the number of bigrassmannian
permutations below a permutation in Bruhat order as Reading
suggested (2002) and afterward the author developed (2011). Second,
bigrassmannian determinant is a q-analog of the determinant with respect
to our statistic. It plays a key role for a determinantal expression
of those polynomials. We further show that bigrassmannian determinant
satisfies weighted condensation as a generalization of Dodgson,
Jacobi-Desnanot and Robbins-Rumsey (1986)
Schubert Numbers
This thesis discusses intersections of the Schubert varieties in the flag variety associated to a vector space of dimension n. The Schubert number is the number of irreducible components of an intersection of Schubert varieties. Our main result gives the lower bound on the maximum of Schubert numbers. This lower bound varies quadratically with n. The known lower bound varied only linearly with n. We also establish a few technical results of independent interest in the combinatorics of strong Bruhat orders
Gravity on an extended brane in six-dimensional warped flux compactifications
We study linearized gravity in a six-dimensional Einstein-Maxwell model of
warped braneworlds, where the extra dimensions are compactified by a magnetic
flux. It is difficult to construct a strict codimension two braneworld with
matter sources other than pure tension. To overcome this problem we replace the
codimension two defect by an extended brane, with one spatial dimension
compactified on a Kaluza-Klein circle. Our background is composed of a warped,
axisymmetric bulk and one or two branes. We find that weak gravity sourced by
arbitrary matter on the brane(s) is described by a four-dimensional
scalar-tensor theory. We show, however, that the scalar mode is suppressed at
long distances and hence four-dimensional Einstein gravity is reproduced on the
brane.Comment: 20 pages, 7 figures; v2: references and comments added; v3: version
published in Physical Review
Ramanujan-Shen's differential equations for Eisenstein series of level 2
Ramanujan (1916) and Shen (1999) discovered differential equations for
classical Eisenstein series. Motivated by them, we derive new differential
equations for Eisenstein series of level 2 from the second kind of Jacobi theta
function. This gives a new characterization of a system of differential
equations by Ablowitz-Chakravarty-Hahn (2006), Hahn (2008), Kaneko-Koike
(2003), Maier (2011) and Toh (2011). As application, we show some arithmetic
results on Ramanujan's tau function.Comment: 21 page
TWO INTEGRAL REPRESENTATIONS FOR APÉRY CONSTANT AND ITS APPLICATIONS TO MULTIPLE ZETA VALUES (Analytic Number Theory and Related Topics)
This article is based on the author's talk "Two integral representations for Apery constant" for Analytic Number Theory and Related Topics, Kyoto RIMS ( online) on October 12, 2021. There are certain overlaps with [9].We generalize the proof of Basel problem by Boo Rim Choe (1987) to obtain two integral representations for Apéry constant. As applications, we also show integral representations for multiple values ζ(3, 2, ... , 2) and t(3, 2, ... , 2)
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