scholarly journals Unique Rectification in $d$-Complete Posets: Towards the $K$-Theory of Kac-Moody Flag Varieties

10.37236/7903 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Rahul Ilango ◽  
Oliver Pechenik ◽  
Michael Zlatin
Keyword(s):  
Large Class ◽  
Cohomology Theory ◽  
Combinatorial Theory ◽  
Flag Varieties ◽  
Jeu De Taquin ◽  
The Family ◽  
Combinatorial Formulas ◽  
Structure Constants ◽  
K Theory

The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of 'unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's '$d$-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of $d$-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain $K$-theoretic Schubert structure constants in the Kac-Moody setting.

scholarly journals Decompositions of Grothendieck Polynomials

2017 ◽  
Vol 2019 (10) ◽  
pp. 3214-3241 ◽  
Author(s):  
Oliver Pechenik ◽  
Dominic Searles
Keyword(s):  
Type A ◽  
General Context ◽  
Flag Varieties ◽  
Combinatorial Formula ◽  
Structure Constants ◽  
Schubert Classes ◽  
K Theory ◽  
Standing Problem ◽  
Grothendieck Polynomials

AbstractWe investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the $K$-theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux–M.-P. Schützenberger (1982) serve as polynomial representatives for $K$-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $n$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $\beta$-Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $K$-theory, and we state our results in this more general context.

scholarly journals Euler characteristics in the quantum K-theory of flag varieties

2020 ◽  
Vol 26 (2) ◽  
Author(s):  
Anders S. Buch ◽  
Sjuvon Chung ◽  
Changzheng Li ◽  
Leonardo C. Mihalcea
Keyword(s):  
Flag Varieties ◽  
Euler Characteristics ◽  
K Theory

scholarly journals Structure constants for K-theory of Grassmannians, revisited

2016 ◽  
Vol 144 ◽  
pp. 306-325 ◽  
Author(s):  
Huilan Li ◽  
Jennifer Morse ◽  
Patrick Shields
Keyword(s):  
Structure Constants ◽  
K Theory

scholarly journals Antipode Formulas for some Combinatorial Hopf Algebras

10.37236/5949 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Rebecca Patrias
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Explicit Formulas ◽  
Quasisymmetric Functions ◽  
Noncommutative Symmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Combinatorial Formulas ◽  
K Theory

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.

Cohomology for Generalized Bredon Coefficient Systems and Higher K-Theory

2012 ◽  
Vol 12 (1) ◽  
pp. 99-113
Author(s):  
Aderemi Kuku
Keyword(s):  
Commutative Ring ◽  
Cohomology Theory ◽  
Discrete Groups ◽  
Finitely Generated ◽  
Bredon Cohomology ◽  
K Theory

AbstractLet be a generalized based category (see definition 1.2). In this paper, we construct a cohomology theory in the category of contravariant functors: where R is a commutative ring with identity, which generalizes Bredon cohomology involving finite, profinite or discrete groups.We also study higher K-theory of the category of finitely generated projective objects in and the category of finitely generated objects in and obtain some finiteness and other results.

K-theory of twisted differential operators on flag varieties

1996 ◽  
Vol 123 (1) ◽  
pp. 377-414
Author(s):  
Martin P. Holland ◽  
Patrick Polo
Keyword(s):  
Differential Operators ◽  
Flag Varieties ◽  
K Theory

Non-Commutative Deformations of C(T2) and K-Theory

1997 ◽  
Vol 08 (05) ◽  
pp. 555-571
Author(s):  
Cristina Cerri
Keyword(s):  
Positive Element ◽  
Crossed Product ◽  
C Algebra ◽  
Basic Properties ◽  
C Algebras ◽  
The Family ◽  
K Theory

For each α ≥ 0, let Bα be the universal C*-algebra generated by unitary elements uα, vα and a self-adjoint hα such that ||hα|| ≤ α and [Formula: see text]. In this work we prove that the family {Bα}α ∈ [0,∞[ extend the family of soft torus with the same basic properties, i.e., the field of C*-algebras {Bα}α ∈ [0,α0] is continuous and each Bα is a crossed product of a C*-algebra homotopically equivalent to C(S1) by Z. We then show that the K-groups of Bα are isomorphic to Z ⊕ Z. Applying results from the theory of rotation algebras we prove that every positive element (n,m) in K0(Bα) satisfies |m|α ≤ 2πn. It follows that these C*-algebras are not all homotopically equivalent to each other, although they have the same K-groups.

scholarly journals MONOPOLES NEAR THE PLANCK SCALE AND UNIFICATION

2003 ◽  
Vol 18 (24) ◽  
pp. 4403-4441 ◽  
Author(s):  
L. V. LAPERASHVILI ◽  
D. A. RYZHIKH ◽  
H. B. NIELSEN
Keyword(s):  
Standard Model ◽  
Dimensional Space ◽  
Planck Scale ◽  
Asymptotic Freedom ◽  
Group Model ◽  
Gauge Groups ◽  
The Family ◽  
The Standard Model ◽  
Dimensional Space Time ◽  
Structure Constants

Considering our (3+1)-dimensional space–time as, in some way, discrete or lattice with a parameter a = λP, where λP is the Planck length, we have investigated the additional contributions of lattice artifact monopoles to beta functions of the renormalization group equations for the running fine structure constants αi(μ) (i = 1,2,3 correspond to the U(1), SU(2) and SU(3) gauge groups of the Standard Model) in the Family Replicated Gauge Group Model (FRGGM) which is an extension of the Standard Model at high energies. It was shown that monopoles have N fam times smaller magnetic charge in FRGGM than in SM (N fam is the number of families in FRGGM). We have estimated also the enlargement of a number of fermions in FRGGM leading to the suppression of the asymptotic freedom in the non-Abelian theory. We have shown that, in contrast to the case of anti-GUT when the FRGGM undergoes the breakdown at μ = μG ~ 1018 GeV , we have the possibility of unification if the FRGGM-breakdown occurs at μG ~ 1014 GeV . By numerical calculations we obtained an example of the unification of all gauge interactions (including gravity) at the scale μ GUT ≈ 1018.4 GeV . We discussed the possibility of [ SU (5)]3 or [ SO (10)]3 (SUSY or not SUSY) unifications.

scholarly journals Monodromy and K-theory of Schubert curves via generalized jeu de taquin

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maria Monks Gillespie ◽  
Jake Levinson
Keyword(s):  
Local Algorithm ◽  
Jeu De Taquin ◽  
Bijective Correspondence ◽  
One Dimensional ◽  
The Real ◽  
Real Locus ◽  
Real Geometry ◽  
International Audience ◽  
K Theory ◽  
Rational Normal Curve

International audience We establish a combinatorial connection between the real geometry and the K-theory of complex Schubert curves Spλ‚q, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map ω on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of RP1, with ω as the monodromy operator.We provide a fast, local algorithm for computing ω without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the K-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the K-theory and real geometry of Spλ‚q.

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