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Search: MSC category 14M15 ( Grassmannians, Schubert varieties, flag manifolds [See also 32M10, 51M35] )

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1. 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

2. 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 $(n-1)\cdot \operatorname{degree}(T(G/P))/n$. 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

3. CMB 2009 (vol 53 pp. 171)

Thomas, Hugh; Yong, Alexander
Multiplicity-Free Schubert Calculus
Multiplicity-free algebraic geometry is the study of subvarieties $Y\subseteq X$ with the ``smallest invariants'' as witnessed by a multiplicity-free 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 multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.

Categories:14M15, 14M05, 05E99

4. 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

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