Selecta Mathematica

Items from 1 to 20 out of 23 results

article

A bijection between Littlewood-Richardson tableaux and rigged configurations

A. N. Kirillov, A. Schilling, M. Shimozono

Selecta Mathematica > 2002 > 8 > 1 > 67-135

. We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials $ K_{\lambda R}(q) $labeled by a partition$ \lambda $and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible$ ...

article

Construction and classification of complex simple Lie algebras via projective geometry

J. M. Landsberg, L. Manivel

Selecta Mathematica > 2002 > 8 > 1 > 137-159

. We construct the complex simple Lie algebras using elementary algebraic geometry. We use our construction to obtain a new proof of the classification of complex simple Lie algebras that does not appeal to the classification of root systems.

article

Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups

L. Ernstrm, S. Yokura

Selecta Mathematica > 2002 > 8 > 1 > 1-25

Abstract. Fulton and MacPherson asked if there exists a bivariant version of the Chern-Schwartz-MacPherson class. Brasselet solved this problem affirmatively in the category of analytic varieties and cellular morphisms. However, it has not been solved in the general case and the uniqueness of such a bivariant Chern class is still open. In this paper we show the unique existence of the bivariant Chern-Schwartz-MacPherson...

article

When is the Fourier transform of an elementary function elementary?

P. Etingof, D. Kazhdan, A. Polishchuk

Selecta Mathematica > 2002 > 8 > 1 > 27-66

Abstract. Let F be a local field, $\psi$ a nontrivial unitary additive character of F, and V a finite dimensional vector space over F. Let us say that a complex function on V is elementary if it has the form $g(x) = C\psi(Q(x))\prod^{k}_{j = 1}\chi_j(P_j(x)), x \in V$ , where $C \in \mathbb{C}$ , Q is a rational function (the phase function), $P_j$ are polynomials, and $\chi_j$ multiplicative characters...

article

Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups

L. Ernström, S. Yokura

Selecta Mathematica > 2002 > 8 > 1 > 1-25

. Fulton and MacPherson asked if there exists a bivariant version of the Chern-Schwartz-MacPherson class. Brasselet solved this problem affirmatively in the category of analytic varieties and cellular morphisms. However, it has not been solved in the general case and the uniqueness of such a bivariant Chern class is still open. In this paper we show the unique existence of the bivariant Chern-Schwartz-MacPherson...

article

A bijection between Littlewood-Richardson tableaux and rigged configurations

A. N. Kirillov, A. Schilling, M. Shimozono

Selecta Mathematica > 2002 > 8 > 1 > 67-135

Abstract. We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials $K_{\lambda R}(q)$ labeled by a partition $\lambda$ and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the...

article

When is the Fourier transform of an elementary function elementary?

P. Etingof, D. Kazhdan, A. Polishchuk

Selecta Mathematica > 2002 > 8 > 1 > 27-66

. Let F be a local field, $\psi$ a nontrivial unitary additive character of F, and V a finite dimensional vector space over F. Let us say that a complex function on V is elementary if it has the form $ g(x) = C\psi(Q(x))\prod^{k}_{j = 1}\chi_j(P_j(x)), x \in V $, where$ C \in \mathbb{C} $ , Q is a rational function (the phase function), $P_j$ are polynomials, and $\chi_j$ multiplicative...

article

Construction and classification of complex simple Lie algebras via projective geometry

J. M. Landsberg, L. Manivel

Selecta Mathematica > 2002 > 8 > 1 > 137-159

Abstract. We construct the complex simple Lie algebras using elementary algebraic geometry. We use our construction to obtain a new proof of the classification of complex simple Lie algebras that does not appeal to the classification of root systems.

article

Mod 2 cohomology of combinatorial Grassmannians

L. Anderson, J.F. Davis

Selecta Mathematica > 2002 > 8 > 2 > 161-200

Abstract. Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization...

article

Cyclic operads and algebra of chord diagrams

V. Hinich, A. Vaintrob

Selecta Mathematica > 2002 > 8 > 2 > 237-282

Abstract. We prove that the algebra $\cal A$ of chord diagrams, the dual to the associated graded algebra of Vassiliev knot invariants, is isomorphic to the universal enveloping algebra of a Casimir Lie algebra in a certain tensor category (the PROP for Casimir Lie algebras). This puts on a firm ground a known statement that the algebra $\cal A$ looks and behaves like a universal enveloping algebra...

article

On the characterization of complex Shimura varieties

Y. Varshavsky

Selecta Mathematica > 2002 > 8 > 2 > 283-314

Abstract. In this paper we recall basic properties of complex Shimura varieties and show that they actually characterize them. This characterization immediately implies the explicit form of Kazhdans theorem on the conjugation of Shimura varieties. It also implies the existence of unique equivariant models over the reflex field of Shimura varieties corresponding to adjoint groups and the existence...

article

Third-order scalar evolution equations with conservation laws

K. Foltinek

Selecta Mathematica > 2002 > 8 > 2 > 201-235

Abstract. Associated to the family of third-order quasilinear scalar evolution equations is the geometry of point transformations. This geometry provides a framework from which to study the structure of conservation laws of the equation, and to study the special nature of the geometry of those equations which do possess conservation laws. There is an easy and obvious normal form for equations which...

article

The rhus integral of rational homology 3-spheres II: Invariance and universality

D. Bar-Natan, S. Garoufalidis, L. Rozansky, D. P. Thurston

Selecta Mathematica > 2002 > 8 > 3 > 341-371

Abstract. We continue the work started in [-I], and prove the invariance and universality in the class of finite type invariants of the object defined and motivated there, namely the rhus integral of rational homology 3-spheres. Our main tool in proving invariance is a translation scheme that translates statements in multi-variable calculus (Gaussian integration, integration by parts, etc.) to statements...

article

Combinatorics of the $\widehat{\mathfrak{sl}}_2$ spaces of coinvariants: loop Heisenberg modules and recursion

B. Feigin, R. Kedem, S. Loktev, T. Miwa, more

Selecta Mathematica > 2002 > 8 > 3 > 419-474

Abstract. The spaces of coinvariants are quotient spaces of integrable $\widehat{\mathfrak{sl}}_2$ modules by subspaces generated by the actions of certain subalgebras labeled by a set of points on a complex line. When all the points are distinct, the spaces of coinvariants essentially coincide with the spaces of conformal blocks in the WZW conformal field theory and their dimensions are given by...

article

Intersection cohomology of Drinfeld‚s compactifications

A. Braverman, M. Finkelberg, D. Gaitsgory, I. Mirković

Selecta Mathematica > 2002 > 8 > 3 > 381-418

. Let X be a smooth complete curve, G be a reductive group and $P \subset G$ a parabolic. Following Drinfeld, one defines a (relative) compactification $\widetilde{\hbox{\rm Bun}\,}_P$ of the moduli stack of P-bundles on X. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of $\widetilde{\hbox{\rm Bun}\,}_P$ . The description is given in terms...

article

The rhus integral of rational homology 3-spheres I: A highly non trivial flat connection on S3

D. Bar-Natan, S. Garoufalidis, L. Rozansky, D. P. Thurston

Selecta Mathematica > 2002 > 8 > 3 > 315-339

Abstract. Path integrals do not really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly non-trivial mathematical theorems and theories. We argue that even though non-trivial flat connections on S3 do not really exist, it is beneficial to dream that one exists...

article

Intersection cohomology of Drinfelds compactifications

A. Braverman, M. Finkelberg, D. Gaitsgory, I. Mirkovi

Selecta Mathematica > 2002 > 8 > 3 > 381-418

Abstract. Let X be a smooth complete curve, G be a reductive group and $P \subset G$ a parabolic. Following Drinfeld, one defines a (relative) compactification $\widetilde{\hbox{\rm Bun}\,}_P$ of the moduli stack of P-bundles on X. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of $\widetilde{\hbox{\rm Bun}\,}_P$ . The description is given in terms...

article

Wonderful blowups associated to group actions

L. A. Borisov, P. E. Gunnells

Selecta Mathematica > 2002 > 8 > 3 > 373-379

Abstract. A group action on a smooth variety provides it with the natural stratification by irreducible components of the fixed point sets of arbitrary subgroups. We show that the corresponding maximal wonderful blowup in the sense of MacPherson-Procesi has only abelian stabilizers. The result is inspired by the abelianization algorithm of Batyrev.

article

Continued fractions, modular symbols, and noncommutative geometry

Y. I. Manin, M. Marcolli

Selecta Mathematica > 2002 > 8 > 3 > 475-521

Abstract. Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols...

article

The Hopf algebra Rep $U_{q} \widehat{\frak g \frak l}_\infty$

E. Frenkel, E. Mukhin

Selecta Mathematica > 2002 > 8 > 4 > 537-635

Abstract. We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of $U_{q} \widehat{\frak g \frak l}_N$ in the limit $N \to \infty$ . The resulting Hopf algebra Rep $U_{q} \widehat{\frak g \frak l}_\infty$ is a tensor product of its Hopf subalgebras $\textrm{Rep}_{a} U_{q} \widehat{\frak g \frak l}_\infty$ , $a \in \mathbb{C}^\times/q^{2{\mathbb Z}}$ ...

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Keywords

3-MANIFOLDS (2)
FINITE TYPE INVARIANTS (2)
GAUSSIAN INTEGRATION (2)
HOLONOMY. (2)
KIRBY MOVES (2)
$\GAMMA$ -FUNCTION (1)
AFFINE LIE ALGEBRA (1)
ALGEBRAIC VARIETY (1)
ALMOST COMPLETE INTEGRABILITY. (1)
BIVARIANT THEORY (1)
CHERN-SCHWARTZ-MACPHERSON CLASS (1)
CHORD DIAGRAMS (1)
COMBINATORICS (1)
CONFORMAL FIELD THEORY. (1)
CONSERVATION LAWS (1)
COVERING SYSTEM. (1)
CYCLIC OPERAD (1)
DIRICHLET SERIES (1)
EVOLUTION EQUATIONS (1)
EVOLUTION EQUATIONS. (1)
FINITE GROUP ACTION (1)
FOURIER TRANSFORM (1)
FROBENIUS HOMOMORPHISMS. (1)
G-STRUCTURES (1)
GAUSS-KUZMIN OPERATOR (1)
GENERALIZED KOSTKA POLYNOMIALS. (1)
HOMOGENEOUS VARIETY (1)
INTERSECTION COHOMOLOGY (1)
INTERSECTION PRODUCT (1)
K-THEORY OF C*-ALGEBRAS. (1)
KEY WORDS. ALGEBRAIC VARIETY, BIVARIANT THEORY, INTERSECTION PRODUCT, CHERN-SCHWARTZ-MACPHERSON CLASS, SPECIALIZATION. (1)
KEY WORDS. FOURIER TRANSFORM, STATIONARY PHASE, LOCAL FIELD, $ \GAMMA $-FUNCTION, COVERING SYSTEM. (1)
KEY WORDS. HOMOGENEOUS VARIETY, LIE ALGEBRA, SECANT VARIETY. (1)
KEY WORDS. LITTLEWOOD-RICHARDSON TABLEAUX, RIGGED CONFIGURATIONS, GENERALIZED KOSTKA POLYNOMIALS. (1)
KEY WORDS. MODULI SPACE OF BUNDLES, INTERSECTION COHOMOLOGY, PLÜCKER RELATIONS. (1)
LIE ALGEBRA (1)
LIMITING MODULAR SYMBOLS (1)
LITTLEWOOD-RICHARDSON TABLEAUX (1)
LOCAL FIELD (1)
MACPHERSONIAN (1)
MATROID BUNDLE. (1)
MODULAR OPERAD (1)
MODULI SPACE OF BUNDLES (1)
NONCOMMUTATIVE MODULAR CURVES (1)
ORIENTED MATROID (1)
PLCKER RELATIONS. (1)
QUANTUM AFFINE GROUPS. (1)
QUANTUM GROUP. (1)
REPRESENTATION THEORY (1)
RIGGED CONFIGURATIONS (1)
SECANT VARIETY. (1)
SHIMURA VARIETIES (1)
SPECIALIZATION. (1)
STABILIZER SUBGROUP. (1)
STATIONARY PHASE (1)
SYMMETRIC PRODUCTS (1)
VASSILIEV INVARIANTS (1)
VECTOR BUNDLE (1)
W-ALGEBRA (1)
WEIL DESCENT. (1)
WONDERFUL BLOWUPS (1)
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INFONA - science communication portal

Selecta Mathematica

A bijection between Littlewood-Richardson tableaux and rigged configurations

Construction and classification of complex simple Lie algebras via projective geometry

Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups

When is the Fourier transform of an elementary function elementary?

Bivariant Chern-Schwartz-MacPherson classes with values in Chow groups

A bijection between Littlewood-Richardson tableaux and rigged configurations

When is the Fourier transform of an elementary function elementary?

Construction and classification of complex simple Lie algebras via projective geometry

Mod 2 cohomology of combinatorial Grassmannians

Cyclic operads and algebra of chord diagrams

On the characterization of complex Shimura varieties

Third-order scalar evolution equations with conservation laws

The rhus integral of rational homology 3-spheres II: Invariance and universality

Combinatorics of the $\widehat{\mathfrak{sl}}_2$ spaces of coinvariants: loop Heisenberg modules and recursion

Intersection cohomology of Drinfeld‚s compactifications

The rhus integral of rational homology 3-spheres I: A highly non trivial flat connection on S3

Intersection cohomology of Drinfelds compactifications

Wonderful blowups associated to group actions

Continued fractions, modular symbols, and noncommutative geometry

The Hopf algebra Rep $U_{q} \widehat{\frak g \frak l}_\infty$

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