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1.
Julianna S. Tymoczko 《Selecta Mathematica, New Series》2007,13(2):353-367
Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety arising in the study of quantum cohomology,
geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent
Hessenberg varieties for all classical types, generalizing results of De Concini–Lusztig–Procesi and Kostant. This paving
is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of the paving
and their dimensions are identified by combinatorial conditions on roots. We use the paving to prove these Hessenberg varieties
have no odd-dimensional homology.
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Frontispiece: Elucidation of the Oxygen Reduction Volcano in Alkaline Media using a Copper–Platinum(111) Alloy
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Representation stability of the cohomology of Springer varieties and some combinatorial consequences
Journal of Algebraic Combinatorics - A sequence of $$S_n$$ -representations $$\{V_n\}$$ is said to be uniformly representation stable if the decomposition of $$V_n = \bigoplus _{\mu } c_{\mu ,n}... 相似文献
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Eric Sommers Julianna Tymoczko 《Transactions of the American Mathematical Society》2006,358(8):3493-3509
Let be a simple algebraic group over the complex numbers containing a Borel subgroup . Given a -stable ideal in the nilradical of the Lie algebra of , we define natural numbers which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types and some other types.
When , we recover the usual exponents of by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.
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Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. We give a “Giambelli formula” expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, we show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. We also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and we give closed combinatorial formulas for the coefficients in many cases. We introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use our results to determine when such schemes are reduced. 相似文献
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Julianna Tymoczko 《Journal of Algebraic Combinatorics》2012,35(4):611-632
Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic
operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and
\mathfrak sl2\mathfrak {sl}_{2}. Building on a result of Kuperberg, Khovanov–Kuperberg found a recursive algorithm giving a bijection between standard Young
tableaux of shape 3×n and irreducible webs for
\mathfraksl3\mathfrak{sl}_{3} whose boundary vertices are all sources. 相似文献
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Geometriae Dedicata - This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each... 相似文献
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