scholarly journals Inverse K-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

2021 ◽  
Vol 9 ◽  
Author(s):  
Takafumi Kouno ◽  
Satoshi Naito ◽  
Daniel Orr ◽  
Daisuke Sagaki
Keyword(s):  
Flag Manifolds ◽  
Line Bundles ◽  
Affine Hecke Algebra ◽  
Double Affine Hecke Algebra ◽  
Bruhat Graph ◽  
Schubert Classes ◽  
Explicit Determination ◽  
Schubert Class ◽  
Quantum Bruhat Graph

Abstract We prove an explicit inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb {Z}\left [q^{\pm 1}\right ]$ -linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars $e^{\lambda }$ , where $\lambda $ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type $E_8$ . The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetric q-Toda operators for minuscule weights in ADE type.

scholarly journals Chern–Schwartz–MacPherson classes for Schubert cells in flag manifolds

2016 ◽  
Vol 152 (12) ◽  
pp. 2603-2625 ◽  
Author(s):  
Paolo Aluffi ◽  
Leonardo C. Mihalcea
Keyword(s):  
Divided Difference ◽  
Flag Manifold ◽  
Flag Manifolds ◽  
Difference Operators ◽  
Generalized Flag Manifold ◽  
Effective Combination ◽  
A Cell ◽  
Schubert Classes ◽  
Dimension One ◽  
Schubert Class

We obtain an algorithm computing the Chern–Schwartz–MacPherson (CSM) classes of Schubert cells in a generalized flag manifold$G/B$. In analogy to how the ordinary divided difference operators act on Schubert classes, each CSM class of a Schubert class is obtained by applying certain Demazure–Lusztig-type operators to the CSM class of a cell of dimension one less. These operators define a representation of the Weyl group on the homology of$G/B$. By functoriality, we deduce algorithmic expressions for CSM classes of Schubert cells in any flag manifold$G/P$. We conjecture that the CSM classes of Schubert cells are an effective combination of (homology) Schubert classes, and prove that this is the case in several classes of examples. We also extend our results and conjecture to the torus equivariant setting.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

Two new measurement methods for explicit determination of complex permittivity

1998 ◽  
Vol 46 (11) ◽  
pp. 1614-1619 ◽  
Author(s):  
C. Wan ◽  
B. Nauwelaers ◽  
W. De Raedt ◽  
M. Van Rossum
Keyword(s):  
Complex Permittivity ◽  
Measurement Methods ◽  
Explicit Determination

scholarly journals Schubert Class and Cyclotomic NilHecke Algebras

2021 ◽  
Vol 28 (03) ◽  
pp. 379-398
Author(s):  
Kai Zhou ◽  
Jun Hu
Keyword(s):  
Algebraic Method ◽  
Basic Algebra ◽  
Graded Algebra ◽  
Algebra Isomorphism ◽  
Type Formula ◽  
Basis Element ◽  
Schubert Classes ◽  
Positive Integers ◽  
Schubert Class

Let [Formula: see text] and [Formula: see text] be positive integers such that [Formula: see text], and let [Formula: see text] be the Grassmannian which consists of the set of [Formula: see text]-dimensional subspaces of [Formula: see text]. There is a [Formula: see text]-graded algebra isomorphism between the cohomology [Formula: see text] of [Formula: see text] and a natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of the type [Formula: see text] cyclotomic nilHecke algebra [Formula: see text]. We show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class [Formula: see text] coincides with the basis element [Formula: see text] constructed by Hu and Liang by purely algebraic method, where [Formula: see text] with [Formula: see text] for each [Formula: see text], and [Formula: see text] is the [Formula: see text]-multipartition of [Formula: see text] associated to [Formula: see text]. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian [Formula: see text] and the [Formula: see text]'s basis ([Formula: see text] is an [Formula: see text]-multipartition of [Formula: see text] with each component being either [Formula: see text] or empty) of the natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of [Formula: see text] is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.

Extensions of I-Bisimple Semigroups

1967 ◽  
Vol 19 ◽  
pp. 419-426 ◽  
Author(s):  
R. J. Warne
Keyword(s):  
Homomorphic Image ◽  
Natural Order ◽  
Isomorphism Theorem ◽  
Maximal Group ◽  
Homomorphism Theorem ◽  
Complete Determination ◽  
Explicit Determination ◽  
Usual Order

A bisimple semigroup S is called I-bisimple if Es, the set of idempotents of S, with its natural order is order-isomorphic to I, the set of integers, under the reverse of the usual order. In (9), the author completely determined the structure of I-bisimple semigroups mod groups; in this paper, he also gave an isomorphism theorem, a homomorphism theorem, an explicit determination of the maximal group homomorphic image, and a complete determination of the congruences for these semigroups.

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