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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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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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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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 S3symmetry 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.

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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
08:56
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
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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
Wellformed formula
Normal (geometry)
Quantum
Metric system
Cue sports
Classical physics
Trail
Polynomial
Quantenkohomologie
Functional (mathematics)
Observational study
Equivalence relation
Causality
Wellformed 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>
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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 ttest
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)
PoissonKlammer
Projective plane
Numerical analysis
Algebra
Basis <Mathematik>
Calculation
Coefficient
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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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[music]