Kazhdan-Lusztig polynomials and drift configurations

The coefficients of the Kazhdan-Lusztig polynomials $P_{v,w}(q)$ are nonnegative integers that are upper semicontinuous on Bruhat order. Conjecturally, the same properties hold for $h$-polynomials $H_{v,w}(q)$ of local rings of Schubert varieties. This suggests a parallel between the two families of polynomials. We prove our conjectures for Grassmannians, and more generally, covexillary Schubert varieties in complete flag varieties, by deriving a combinatorial formula for $H_{v,w}(q)$. We introduce \emph{drift configurations} to formulate a new and compatible combinatorial rule for $P_{v,w}(q)$. From our rules we deduce, for these cases, the coefficient-wise inequality $P_{v,w}(q)\preceq H_{v,w}(q)$.Comment: 26 pages. To appear in Algebra & Number Theor

Distance and intersection number in the curve graph of a surface

In this work, we study the cellular decomposition of $S$ induced by a filling pair of curves $v$ and $w$, $Dec_{v,w}(S) = S - (v \cup w)$, and its connection to the distance function $d(v,w)$ in the curve graph of a closed orientable surface $S$ of genus $g$. Efficient geodesics were introduced by the first author in joint work with Margalit and Menasco in 2016, giving an algorithm that begins with a pair of non-separating filling curves that determine vertices $(v,w)$ in the curve graph of a closed orientable surface $S$ and computing from them a finite set of {\it efficient} geodesics. We extend the tools of efficient geodesics to study the relationship between distance $d(v,w)$, intersection number $i(v,w)$, and $Dec_{v,w}(S)$. The main result is the development and analysis of particular configurations of rectangles in $Dec_{v,w}(S)$ called \textit{spirals}. We are able to show that, in some special cases, the efficient geodesic algorithm can be used to build an algorithm that reduces $i(v,w)$ while preserving $d(v,w)$. At the end of the paper, we note a connection of our work to the notion of extending geodesics.Comment: 20 pages, 17 figures. Changes: A key lemma (Lemma 5.6) was revised to be more precise, an irrelevant proposition (Proposition 2.1) and example were removed, unnecessary background material was taken out, some of the definitions and cited results were clarified (including added figures,) and Proposition 5.7 and Theorem 5.8 have been merged into a single theorem, Theorem 4.

V,W and X in Technicolour Models

Light techni-fermions and pseudo Goldstone bosons that contribute to the electroweak radiative correction parameters V,W and X may relax the constraints on technicolour models from the experimental values of the parameters S and T. Order of magnitude estimates of the contributions to V,W and X from light techni-leptons are made when the the techni-neutrino has a small Dirac mass or a large Majorana mass. The contributions to V,W and X from pseudo Goldstone bosons are calculated in a gauged chiral Lagrangian. Estimates of V,W and X in one family technicolour models suggest that the upper bounds on S and T should be relaxed by between 0.1 and 1 depending upon the precise particle spectrum.Comment: 19 pages + 2 pages of ps figs, SWAT/1

Variations on twists of tuples of hyperelliptic curves and related results

Let f\in\Q[x] be a square-free polynomial of degree $\geq 3$ and $m\geq 3$ be an odd positive integer. Based on our earlier investigations we prove that there exists a function D_{1}\in\Q(u,v,w) such that the Jacobians of the curves \begin{equation*} C_{1}:\;D_{1}y^2=f(x),\quad C_{2}:\;y^2=D_{1}x^m+b,\quad C_{3}:\;y^2=D_{1}x^m+c, \end{equation*} have all positive ranks over \Q(u,v,w). Similarly, we prove that there exists a function D_{2}\in\Q(u,v,w) such that the Jacobians of the curves \begin{equation*} C_{1}:\;D_{2}y^2=h(x),\quad C_{2}:\;y^2=D_{2}x^m+b,\quad C_{3}:\;y^2=x^m+cD_{2}, \end{equation*} have all positive ranks over \Q(u,v,w). Moreover, if $f(x)=x^m+a$ for some $a\in\Z\setminus\{0\}$, we prove the existence of a function D_{3}\in\Q(u,v,w) such that the Jacobians of the curves \begin{equation*} C_{1}:\;y^2=D_{3}x^{m}+a,\quad C_{2}:\;y^2=D_{3}x^m+b,\quad C_{3}:\;y^2=x^m+cD_{3}, \end{equation*} have all positive ranks over \Q(u,v,w). We present also some applications of these results. Finally, we present some results concerning the torsion parts of the Jacobians of the superelliptic curves $y^p=x^{m}(x+a)$ and $y^p=x^{m}(a-x)^{k}$ for a prime $p$ and $0<m<p-2$ and $k<p$ and apply our result in order to prove the existence of a function D\in\Q(u,v,w,t) such that the Jacobians of the curves \begin{equation*} C_{1}:\;Dy^p=x^m(x+a),\quad Dy^p=x^m(x+b) \end{equation*} have both positive rank over \Q(u,v,w,t).Comment: Revised version will appear in Journal of Number Theor

First and second cohomologies of grading-restricted vertex algebras

Let $V$ be a grading-restricted vertex algebra and $W$ a $V$-module. We show that for any $m\in \mathbb{Z}_{+}$, the first cohomology $H^{1}_{m}(V, W)$ of $V$ with coefficients in $W$ introduced by the author is linearly isomorphic to the space of derivations from $V$ to $W$. In particular, $H^{1}_{m}(V, W)$ for $m\in \mathbb{N}$ are equal (and can be denoted using the same notation $H^{1}(V, W)$). We also show that the second cohomology $H^{2}_{\frac{1}{2}}(V, W)$ of $V$ with coefficients in $W$ introduced by the author corresponds bijectively to the set of equivalence classes of square-zero extensions of $V$ by $W$. In the case that $W=V$, we show that the second cohomology $H^{2}_{\frac{1}{2}}(V, V)$ corresponds bijectively to the set of equivalence classes of first order deformations of $V$.Comment: 24 pages. Final version to appear in Communications in Mathematical Physic