1. CMB 2017 (vol 60 pp. 478)
 Carrell, Jim; Kaveh, Kiumars

Springer's Weyl Group Representation via Localization
Let $G$ denote a reductive algebraic group over
$\mathbb{C}$
and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer
variety $\mathcal{B}_x$
is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing
the
Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable
property that
the Weyl group $W$ of $G$ admits a representation on the cohomology
of $\mathcal{B}_x$
even though $W$ rarely acts on $\mathcal{B}_x$ itself. Wellknown constructions
of this action
due to Springer et al use technical machinery from algebraic
geometry.
The purpose of this note is to describe an elementary approach
that gives this action
when $x$ is what we call parabolicsurjective. The idea is to
use localization to construct an action of $W$ on
the equivariant cohomology algebra $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic
subtorus of $G$.
This action descends to $H^*(\mathcal{B}_x)$ via the forgetful map and
gives the desired representation. The parabolicsurjective case
includes all nilpotents of type $A$ and,
more generally, all nilpotents for which it is known that $W$
acts on
$H_S^*(\mathcal{B}_x)$ for some torus $S$.
Our result is deduced from a general theorem describing
when a group action on the cohomology of the fixed point set of a
torus action
on a space lifts to the full cohomology algebra of the space.
Keywords:Springer variety, Weyl group action, equivariant cohomology Categories:14M15, 14F43, 55N91 

2. CMB 2017 (vol 60 pp. 747)
 Huang, Yanhe; Sottile, Frank; Zelenko, Igor

Injectivity of Generalized Wronski Maps
We study linear projections on PlÃ¼cker space whose restriction
to the Grassmannian is a nontrivial branched
cover.
When an automorphism of the Grassmannian preserves the fibers,
we show that the Grassmannian is necessarily
of $m$dimensional linear subspaces in a symplectic vector
space of dimension $2m$, and the linear map is
the Lagrangian involution.
The Wronski map for a selfadjoint linear differential operator
and pole placement map for
symmetric linear systems are natural examples.
Keywords:Wronski map, PlÃ¼cker embedding, curves in Lagrangian Grassmannian, selfadjoint linear differential operator, symmetric linear control system, pole placement map Categories:14M15, 34A30, 93B55 

3. CMB 2010 (vol 53 pp. 757)
 Woo, Alexander

Interval Pattern Avoidance for Arbitrary Root Systems
We extend the idea of interval pattern avoidance defined by Yong and
the author for $S_n$ to arbitrary Weyl groups using the definition of
pattern avoidance due to Billey and Braden, and Billey and Postnikov.
We show that, as previously shown by Yong and the
author for $\operatorname{GL}_n$, interval pattern avoidance is a universal tool for
characterizing which Schubert varieties have certain local properties,
and where these local properties hold.
Categories:14M15, 05E15 

4. CMB 2009 (vol 53 pp. 218)
 Biswas, Indranil

Restriction of the Tangent Bundle of $G/P$ to a Hypersurface
Let P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over $\mathbb C$, such that $n := \dim_{\mathbb C} G/P \geq 4$. Let $\iota \colon Z \hookrightarrow G/P$ be a reduced smooth hypersurface of degree at least $(n1)\cdot \operatorname{degree}(T(G/P))/n$. We prove that the restriction of the tangent bundle $\iota^*TG/P$ is semistable.
Keywords:tangent bundle, homogeneous space, semistability, hypersurface Categories:14F05, 14J60, 14M15 

5. CMB 2009 (vol 53 pp. 171)
 Thomas, Hugh; Yong, Alexander

MultiplicityFree Schubert Calculus
Multiplicityfree algebraic geometry is the study of subvarieties
$Y\subseteq X$ with the ``smallest invariants'' as witnessed by a
multiplicityfree Chow ring decomposition of
$[Y]\in A^{\star}(X)$ into a predetermined
linear basis.
This paper concerns the case of Richardson subvarieties of the Grassmannian
in terms of the Schubert basis. We give a nonrecursive combinatorial
classification of multiplicityfree Richardson varieties, i.e.,
we classify multiplicityfree products of Schubert classes. This answers
a question of W. Fulton.
Categories:14M15, 14M05, 05E99 

6. CMB 2009 (vol 52 pp. 200)
 Gatto, Letterio; Santiago, Ta\'\i se

Schubert Calculus on a Grassmann Algebra
The ({\em classical}, {\em small quantum}, {\em equivariant})
cohomology ring of the grassmannian $G(k,n)$ is generated by
certain derivations operating on an exterior algebra of a free
module of rank $n$ ( Schubert calculus on a Grassmann
algebra). Our main result gives, in a unified way, a presentation
of all such cohomology rings in terms of generators and
relations. Using results of Laksov and Thorup, it also provides
a presentation of the universal
factorization algebra of a monic polynomial of degree $n$ into the
product of two monic polynomials, one of degree $k$.
Categories:14N15, 14M15 
