A quotient of the ring of symmetricfunctions generalizing quantum cohomology

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Video in TIB AV-Portal: A quotient of the ring of symmetricfunctions generalizing quantum cohomology

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A quotient of the ring of symmetricfunctions generalizing quantum cohomology
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Quotients of Symmetric Polynomial Rings Deforming the Cohomology of the Grassmannian
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2021
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English

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Abstract
One of the many connections between Grassmannians and combinatorics is cohomological: The cohomology ring of a Grassmannian Gr(k,n) is a quotient of the ring S of symmetric polynomials in k variables. More precisely, it is the quotient of S by the ideal generated by the k consecutive complete homogeneous symmetric polynomials hn−k,hn−k+1,…,hn. We deform this quotient, by replacing the ideal by the ideal generated by hn−k−a1,hn−k+1−a2,…,hn−ak for some k fixed elements a1,a2,…,ak of the base ring. This generalizes both the classical and the quantum cohomology rings of Gr(k,n). We find three bases for the new quotient, as well as an S3-symmetry of its structure constants, a “rim hook rule” for straightening arbitrary Schur polynomials, and a fairly complicated Pieri rule. We conjecture that the structure constants are nonnegative in an appropriate sense (treating the ai as signed indeterminate), which suggests a geometric or combinatorial meaning for the quotient.
Commutative property Presentation of a group Homologie Multiplication sign Calculus Parameter (computer programming) Mereology Perspective (visual) Invariant (mathematics) Negative number Ranking Damping Grassmannian Extension (kinesiology) James Waddell Alexander II Algebra Multiplication Modulo (jargon) Cohomology Geometry Logical constant Polynomial Computability Staff (military) Mereology Price index Variable (mathematics) Complete metric space Permutation Hand fan Partition (number theory) Category of being Ring (mathematics) Chain Theorem Quantum Right angle Ideal (ethics) Cycle (graph theory) Figurate number Metric system Spacetime Classical physics Polynomial Quantenkohomologie Free group Module (mathematics) Observational study Algebraic structure Infinity Affine space Element (mathematics) Sequence Complex number Profil (magazine) Term (mathematics) Ring (mathematics) Queue (abstract data type) Modulform Quantum Set theory Noise (electronics) Standard deviation Variety (linguistics) Forcing (mathematics) Classical physics Physical law Model theory Rectangle Numerical analysis Film editing Basis <Mathematik> Factory (trading post) Musical ensemble Game theory Table (information) Diagram Family Flag
Axiom of choice Logical constant Complex (psychology) Rifling Group action Correspondence (mathematics) Price index Invariant (mathematics) Different (Kate Ryan album) Negative number Hill differential equation Social class Geometry Gamma function Propositional formula Staff (military) Maxima and minima Sequence Permutation Product (business) Partition (number theory) Wave Symmetric matrix Ring (mathematics) Symmetry (physics) Order (biology) Theorem Moving average Cycle (graph theory) Spacetime Point (geometry) Free group Module (mathematics) Wage labour Tournament (medieval) Axonometric projection Product (business) Frequency Coefficient Term (mathematics) Ring (mathematics) Modulform Maß <Mathematik> Surjective function Alpha (investment) Quantum state Content (media) Heat transfer Basis <Mathematik> Line (geometry) Mortality rate Cartesian coordinate system Rectangle Sign (mathematics) Spring (hydrology) Social class Game theory Musical ensemble Table (information) Integer Matrix (mathematics) Standard deviation Beta function Multiplication sign Sheaf (mathematics) Expandierender Graph Mereology Grothendieck topology Permutation Analogy Square number Cuboid Extension (kinesiology) Determinant Cohomology Triangle Polynomial Process (computing) Special unitary group Price index Variable (mathematics) Element (mathematics) Tessellation Well-formed formula Normal (geometry) Quantum Metric system Cue sports Classical physics Trail Polynomial Quantenkohomologie Functional (mathematics) Observational study Equivalence relation Causality Well-formed formula Reduction of order Curve fitting Linear map Module (mathematics) Noise (electronics) Dependent and independent variables Military base Forcing (mathematics) Horizon Numerical analysis Diameter Film editing Algebra Friction Basis <Mathematik> LAN party Coefficient Diagram Maß <Mathematik>
Logical constant Group action Presentation of a group Analogy Expected value Connected space Sign (mathematics) Invariant (mathematics) Different (Kate Ryan album) Negative number Hill differential equation Proof theory Smith chart Parameter (computer programming) Maxima and minima Sequence Permutation Hand fan Partition (number theory) Category of being Arithmetic mean Symmetric matrix Ring (mathematics) Bernstein polynomial Phase transition Order (biology) Chain Theorem Moving average Cycle (graph theory) Spacetime Slide rule Beat (acoustics) Module (mathematics) Student's t-test Infinity Product (business) Scheduling (computing) Element (mathematics) Coefficient Term (mathematics) Ring (mathematics) Manifold Äquivariante Abbildung Modulform Physical constant Matching (graph theory) Chemical equation Weight Independence (probability theory) Basis <Mathematik> Group action Rectangle Mathematics Spring (hydrology) Function (mathematics) Set theory Musical ensemble Game theory Family Standard deviation Primality test Length Multiplication sign Parameter (computer programming) Mereology Grothendieck topology Duality (mathematics) Bernstein polynomial Flag Circle Körper <Algebra> Logical constant Polynomial Process (computing) Computability Term (mathematics) Variable (mathematics) Element (mathematics) Ideal (ethics) Asymmetry Metric system Thomas Bayes Cue sports Polynomial Trail Functional (mathematics) Algebraic structure Inclined plane Power (physics) 2 (number) Local Group Sequence Regular graph Quotient Green's function Harmonic analysis Set theory Hydraulic jump Condition number Module (mathematics) Multiplication Military base Forcing (mathematics) Poisson-Klammer Projective plane Numerical analysis Algebra Basis <Mathematik> Calculation Coefficient
Group action Beta function Calculus Parameter (computer programming) Function (mathematics) Invariant (mathematics) Forest Grassmannian James Waddell Alexander II Multiplication Cohomology Frobenius method Geometry Polynomial Logical constant Propositional formula Parameter (computer programming) Staff (military) Maxima and minima Mereology Element (mathematics) Product (business) Degree (graph theory) Partition (number theory) Symmetric matrix Chain Theorem Quantum Ideal (ethics) Figurate number Quantenkohomologie Module (mathematics) Divisor Algebraic structure Coordinate system Equivalence relation Affine space Product (business) Goodness of fit Coefficient Ring (mathematics) Selectivity (electronic) Quantum Linear map Time zone Matching (graph theory) Variety (linguistics) Classical physics Physical law Basis <Mathematik> Braid Existence Numerical analysis Basis <Mathematik> Set theory Musical ensemble Coefficient Integer Flag
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