Drexel University

A quotient of the ring of symmetric functions generalizing quantum cohomology

Darij Grinberg

28 January 2019

Drexel University, Philadelphia, PA

slides: http:

//www.cip.ifi.lmu.de/~grinberg/algebra/drexel2019.pdf paper: http:

//www.cip.ifi.lmu.de/~grinberg/algebra/basisquot.pdf overview: http:

//www.cip.ifi.lmu.de/~grinberg/algebra/fpsac19.pdf

1 / 31

What is this about?

From a modern point of view, Schubert calculus (a.k.a.

classical enumerative geometry, or Hilbert’s 15th problem) is about two cohomology rings:

H^∗



 Gr (k,n)

| {z }

Grassmannian



 and H^∗





 Fl (n)

| {z }

flag variety







(both varieties over C).

In this talk, we are concerned with the first.

Classical result: as rings, H^∗(Gr (k,n))

∼= (symmetric polynomials inx1,x2, . . . ,xk overZ) (hn−k+1,hn−k+2, . . . ,h_n)_ideal,

where the h_i are complete homogeneous symmetric polynomials (to be defined soon).

2 / 31

What is this about?

From a modern point of view, Schubert calculus (a.k.a.

classical enumerative geometry, or Hilbert’s 15th problem) is about two cohomology rings:

H^∗



 Gr (k,n)

| {z }

Grassmannian



 and H^∗





 Fl (n)

| {z }

flag variety







(both varieties over C).

In this talk, we are concerned with the first.

Classical result: as rings, H^∗(Gr (k,n))

∼= (symmetric polynomials inx1,x2, . . . ,xk overZ) (hn−k+1,hn−k+2, . . . ,h_n)_ideal,

where the h_i are complete homogeneous symmetric polynomials (to be defined soon).

2 / 31

What is this about?

From a modern point of view, Schubert calculus (a.k.a.

classical enumerative geometry, or Hilbert’s 15th problem) is about two cohomology rings:

H^∗



 Gr (k,n)

| {z }

Grassmannian



 and H^∗





 Fl (n)

| {z }

flag variety







(both varieties over C).

In this talk, we are concerned with the first.

Classical result: as rings, H^∗(Gr (k,n))

∼= (symmetric polynomials inx1,x2, . . . ,xk overZ) (hn−k+1,hn−k+2, . . . ,h_n)_ideal,

where the h_i are complete homogeneous symmetric polynomials (to be defined soon).

2 / 31

Quantum cohomology of Gr(k,n)

(Small)Quantum cohomology is a deformation of

cohomology from the 1980–90s. For the Grassmannian, it is QH^∗(Gr (k,n))

∼= (symmetric polynomials in x₁,x₂, . . . ,x_k overZ[q])

hn−k+1,hn−k+2, . . . ,hn−1,h_n+ (−1)^kq

ideal. Many properties of classical cohomology still hold here.

In particular: QH^∗(Gr (k,n)) has a Z[q]-module basis (sλ)_λ∈P

k,n of (projected) Schur polynomials (to be defined soon), with λranging over all partitions with ≤k parts and each part≤n−k. The structure constants are the

Gromov–Witten invariants. References:

Aaron Bertram, Ionut Ciocan-Fontanine, William Fulton, Quantum multiplication of Schur polynomials, 1999.

Alexander Postnikov, Affine approach to quantum Schubert calculus, 2005.

3 / 31

Quantum cohomology of Gr(k,n)

(Small)Quantum cohomology is a deformation of

cohomology from the 1980–90s. For the Grassmannian, it is QH^∗(Gr (k,n))

∼= (symmetric polynomials in x₁,x₂, . . . ,x_k overZ[q])

hn−k+1,hn−k+2, . . . ,hn−1,h_n+ (−1)^kq

ideal. Many properties of classical cohomology still hold here.

In particular: QH^∗(Gr (k,n)) has a Z[q]-module basis (sλ)_λ∈P

k,n of (projected) Schur polynomials (to be defined soon), with λranging over all partitions with ≤k parts and each part≤n−k. The structure constants are the

Gromov–Witten invariants. References:

Aaron Bertram, Ionut Ciocan-Fontanine, William Fulton, Quantum multiplication of Schur polynomials, 1999.

Alexander Postnikov, Affine approach to quantum Schubert calculus, 2005.

3 / 31

Where are we going?

Goal: Deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

The new ring has no geometric interpretation known so far, but various properties suggesting such an interpretation likely exists.

I will now start from scratch and define standard notations around symmetric polynomials, then introduce the deformed cohomology ring algebraically.

4 / 31

Where are we going?

Goal: Deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

The new ring has no geometric interpretation known so far, but various properties suggesting such an interpretation likely exists.

I will now start from scratch and define standard notations around symmetric polynomials, then introduce the deformed cohomology ring algebraically.

4 / 31

Where are we going?

Goal: Deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

The new ring has no geometric interpretation known so far, but various properties suggesting such an interpretation likely exists.

I will now start from scratch and define standard notations around symmetric polynomials, then introduce the deformed cohomology ring algebraically.

4 / 31

A more general setting: P and S Let kbe a commutative ring.

Let N={0,1,2, . . .}. Let k∈N.

Let P =k[x1,x2, . . . ,xk] be the polynomial ring ink indeterminates overk.

For eachk-tuple α∈N^k and each i ∈ {1,2, . . . ,k}, let αi be thei-th entry ofα. Same for infinite sequences.

For eachα∈N^k, letx^α be the monomial x₁^α1x₂^α2· · ·x_k^αk, and let |α|be the degreeα₁+α₂+· · ·+α_k of this monomial. Let S denote the ring of symmetricpolynomials inP. These are the polynomials f ∈ P satisfying

f (x1,x2, . . . ,xk) =f x_σ(1),x_σ(2), . . . ,x_σ(k) for all permutationsσ of{1,2, . . . ,k}.

Theorem (Artin ≤1944): TheS-moduleP is free with basis (x^α)_α∈Nk;αi<i for eachi (or, informally: “

x₁^<1x₂^<2· · ·x_k^<k

”). Example: Fork = 3, this basis is 1,x₃,x₃²,x₂,x₂x₃,x₂x₃²

.

5 / 31

A more general setting: P and S Let kbe a commutative ring.

Let N={0,1,2, . . .}. Let k∈N.

Let P =k[x1,x2, . . . ,xk] be the polynomial ring ink indeterminates overk.

For eachk-tupleα ∈N^k and each i ∈ {1,2, . . . ,k}, let αi be thei-th entry ofα. Same for infinite sequences.

For eachα∈N^k, letx^α be the monomial x₁^α1x₂^α2· · ·x_k^αk, and let |α|be the degreeα₁+α₂+· · ·+α_k of this monomial. Let S denote the ring of symmetricpolynomials inP. These are the polynomials f ∈ P satisfying

f (x1,x2, . . . ,xk) =f x_σ(1),x_σ(2), . . . ,x_σ(k) for all permutationsσ of{1,2, . . . ,k}.

Theorem (Artin ≤1944): TheS-moduleP is free with basis (x^α)_α∈Nk;αi<i for eachi (or, informally: “

x₁^<1x₂^<2· · ·x_k^<k

”). Example: Fork = 3, this basis is 1,x₃,x₃²,x₂,x₂x₃,x₂x₃²

.

5 / 31

A more general setting: P and S Let kbe a commutative ring.

Let N={0,1,2, . . .}. Let k∈N.

Let P =k[x1,x2, . . . ,xk] be the polynomial ring ink indeterminates overk.

For eachk-tupleα ∈N^k and each i ∈ {1,2, . . . ,k}, let αi be thei-th entry ofα. Same for infinite sequences.

For eachα∈N^k, let x^α be the monomial x₁^α1x₂^α2· · ·x_k^αk, and let |α|be the degreeα₁+α₂+· · ·+α_k of this monomial.

Let S denote the ring of symmetricpolynomials inP. These are the polynomials f ∈ P satisfying

f (x1,x2, . . . ,xk) =f x_σ(1),x_σ(2), . . . ,x_σ(k) for all permutationsσ of{1,2, . . . ,k}.

Theorem (Artin ≤1944): TheS-moduleP is free with basis (x^α)_α∈Nk;αi<i for eachi (or, informally: “

x₁^<1x₂^<2· · ·x_k^<k

”). Example: Fork = 3, this basis is 1,x₃,x₃²,x₂,x₂x₃,x₂x₃²

.

5 / 31

A more general setting: P and S Let kbe a commutative ring.

Let N={0,1,2, . . .}. Let k∈N.

Let P =k[x1,x2, . . . ,xk] be the polynomial ring ink indeterminates overk.

For eachk-tupleα ∈N^k and each i ∈ {1,2, . . . ,k}, let αi be thei-th entry ofα. Same for infinite sequences.

For eachα∈N^k, let x^α be the monomial x₁^α1x₂^α2· · ·x_k^αk, and let |α|be the degreeα₁+α₂+· · ·+α_k of this monomial.

Let S denote the ring of symmetricpolynomials inP. These are the polynomials f ∈ P satisfying

f (x1,x2, . . . ,x_k) =f x_σ(1),x_σ(2), . . . ,x_σ(k) for all permutationsσ of{1,2, . . . ,k}.

Theorem (Artin ≤1944): TheS-moduleP is free with basis (x^α)_α∈Nk;αi<i for eachi (or, informally: “

x₁^<1x₂^<2· · ·x_k^<k

”). Example: Fork = 3, this basis is 1,x₃,x₃²,x₂,x₂x₃,x₂x₃²

.

5 / 31

A more general setting: P and S Let kbe a commutative ring.

Let N={0,1,2, . . .}. Let k∈N.

Let P =k[x1,x2, . . . ,xk] be the polynomial ring ink indeterminates overk.

For eachk-tupleα ∈N^k and each i ∈ {1,2, . . . ,k}, let αi be thei-th entry ofα. Same for infinite sequences.

For eachα∈N^k, let x^α be the monomial x₁^α1x₂^α2· · ·x_k^αk, and let |α|be the degreeα₁+α₂+· · ·+α_k of this monomial.

Let S denote the ring of symmetricpolynomials inP. These are the polynomials f ∈ P satisfying

f (x1,x2, . . . ,x_k) =f x_σ(1),x_σ(2), . . . ,x_σ(k) for all permutationsσ of{1,2, . . . ,k}.

Theorem (Artin ≤1944): TheS-moduleP is free with basis (x^α)_α∈Nk;αi<i for eachi (or, informally: “

x₁^<1x₂^<2· · ·x_k^<k

”).

Example: Fork = 3, this basis is 1,x₃,x₃²,x₂,x₂x₃,x₂x₃² .

5 / 31

A more general setting: P and S Let kbe a commutative ring.

Let N={0,1,2, . . .}. Let k∈N.

Let P =k[x1,x2, . . . ,xk] be the polynomial ring ink indeterminates overk.

For eachk-tupleα ∈N^k and each i ∈ {1,2, . . . ,k}, let αi be thei-th entry ofα. Same for infinite sequences.

For eachα∈N^k, let x^α be the monomial x₁^α1x₂^α2· · ·x_k^αk, and let |α|be the degreeα₁+α₂+· · ·+α_k of this monomial.

Let S denote the ring of symmetricpolynomials inP. These are the polynomials f ∈ P satisfying

f (x1,x2, . . . ,x_k) =f x_σ(1),x_σ(2), . . . ,x_σ(k) for all permutationsσ of{1,2, . . . ,k}.

Theorem (Artin ≤1944): TheS-moduleP is free with basis (x^α)_α∈Nk;αi<i for eachi (or, informally: “

x₁^<1x₂^<2· · ·x_k^<k

”).

Example: Fork = 3, this basis is 1,x₃,x₃²,x₂,x₂x₃,x₂x₃² .

5 / 31

A more general setting: P and S Let kbe a commutative ring.

Let N={0,1,2, . . .}. Let k∈N.

Let P =k[x1,x2, . . . ,xk] be the polynomial ring ink indeterminates overk.

For eachk-tupleα ∈N^k and each i ∈ {1,2, . . . ,k}, let αi be thei-th entry ofα. Same for infinite sequences.

For eachα∈N^k, let x^α be the monomial x₁^α1x₂^α2· · ·x_k^αk, and let |α|be the degreeα₁+α₂+· · ·+α_k of this monomial.

Let S denote the ring of symmetricpolynomials inP. These are the polynomials f ∈ P satisfying

f (x1,x2, . . . ,x_k) =f x_σ(1),x_σ(2), . . . ,x_σ(k) for all permutationsσ of{1,2, . . . ,k}.

Theorem (Artin ≤1944): TheS-moduleP is free with basis (x^α)_α∈Nk;αi<i for eachi (or, informally: “

x₁^<1x₂^<2· · ·x_k^<k

”).

Example: Fork = 3, this basis is 1,x₃,x₃²,x₂,x₂x₃,x₂x₃² .

5 / 31

Symmetric polynomials

The ring S of symmetric polynomials inP =k[x1,x2, . . . ,xk] has several bases, usually indexed by certain sets of (integer) partitions.

First, let us recall what partitions are:

6 / 31

k-partitions: definition

A partitionmeans aweakly decreasing sequence of nonnegative integers that has only finitely many nonzero entries.

Ifλ∈N^k is a k-partition, then its Young diagram Y(λ)is defined as a table made out of k left-aligned rows, where the i-th row hasλ_i boxes.

Example: Ifk = 6 andλ= (5,5,3,2,0,0), then

Y (λ) = .

(Empty rows are invisible.)

The same convention applies to partitions.

7 / 31

k-partitions: definition

A partitionmeans aweakly decreasing sequence of nonnegative integers that has only finitely many nonzero entries.

Examples: (4,2,2,0,0,0, . . .) and (3,2,0,0,0,0, . . .) and (5,0,0,0,0,0, . . .) are three partitions.

(2,3,2,0,0,0, . . .) and (2,1,1,1, . . .) are not.

Ifλ∈N^k is a k-partition, then its Young diagram Y(λ)is defined as a table made out of k left-aligned rows, where the i-th row hasλ_i boxes.

Example: Ifk = 6 andλ= (5,5,3,2,0,0), then

Y (λ) = .

(Empty rows are invisible.)

The same convention applies to partitions.

7 / 31

k-partitions: definition

A k-partitionmeans aweakly decreasingk-tuple (λ1, λ2, . . . , λk)∈N^k.

Ifλ∈N^k is a k-partition, then its Young diagram Y(λ)is defined as a table made out of k left-aligned rows, where the i-th row hasλi boxes.

Example: Ifk = 6 andλ= (5,5,3,2,0,0), then

Y (λ) = .

(Empty rows are invisible.)

The same convention applies to partitions.

7 / 31

k-partitions: definition

A k-partitionmeans aweakly decreasingk-tuple (λ1, λ2, . . . , λk)∈N^k.

Examples: (4,2,2) and (3,2,0) and (5,0,0) are three 3-partitions.

(2,3,2) is not.

Ifλ∈N^k is a k-partition, then its Young diagram Y(λ)is defined as a table made out of k left-aligned rows, where the i-th row hasλi boxes.

Example: Ifk = 6 andλ= (5,5,3,2,0,0), then

Y (λ) = .

(Empty rows are invisible.)

The same convention applies to partitions.

7 / 31

k-partitions: definition

A k-partitionmeans aweakly decreasingk-tuple (λ1, λ2, . . . , λk)∈N^k.

Examples: (4,2,2) and (3,2,0) and (5,0,0) are three 3-partitions.

(2,3,2) is not.

Thus there is a bijection

{k-partitions} → {partitions with at mostk nonzero entries}, λ7→(λ1, λ2, . . . , λk,0,0,0, . . .).

Ifλ∈N^k is a k-partition, then its Young diagram Y(λ) is defined as a table made out of k left-aligned rows, where the i-th row hasλi boxes.

Example: Ifk = 6 andλ= (5,5,3,2,0,0), then

Y (λ) = .

(Empty rows are invisible.)

The same convention applies to partitions.

7 / 31

k-partitions: definition

A k-partitionmeans aweakly decreasingk-tuple (λ1, λ2, . . . , λk)∈N^k.

Examples: (4,2,2) and (3,2,0) and (5,0,0) are three 3-partitions.

(2,3,2) is not.

Ifλ∈N^k is a k-partition, then its Young diagram Y(λ)is defined as a table made out of k left-aligned rows, where the i-th row hasλi boxes.

Example: Ifk = 6 andλ= (5,5,3,2,0,0), then

Y (λ) = .

(Empty rows are invisible.)

The same convention applies to partitions.

7 / 31

k-partitions: definition

A k-partitionmeans aweakly decreasingk-tuple (λ1, λ2, . . . , λk)∈N^k.

Examples: (4,2,2) and (3,2,0) and (5,0,0) are three 3-partitions.

(2,3,2) is not.

Ifλ∈N^k is a k-partition, then its Young diagram Y(λ)is defined as a table made out of k left-aligned rows, where the i-th row hasλi boxes.

Example: Ifk = 6 andλ= (5,5,3,2,0,0), then

Y (λ) = .

(Empty rows are invisible.)

The same convention applies to partitions.

7 / 31

Symmetric polynomials: the e-basis

For eachm∈Z, we lete_m denote them-thelementary symmetric polynomial:

em= X

1≤i₁<i2<···<im≤k

xi1xi2· · ·xim = X

α∈{0,1}^k;

|α|=m

x^α ∈ S.

(Thus, e0 = 1, and em= 0 when m<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

e_ν =e_ν1e_ν2· · ·e_ν` ∈ S.

Theorem (Gauss): The commutativek-algebraS is freely generated by the elementary symmetric polynomials

e1,e2, . . . ,ek. (That is, it is generated by them, and they are algebraically independent.)

Equivalent restatement: (eλ)_λis a partition whose entries are≤k

is a basis of the k-module S. Note thatem= 0 when m>k.

8 / 31

Symmetric polynomials: the e-basis

For eachm∈Z, we lete_m denote them-thelementary symmetric polynomial:

em= X

1≤i₁<i2<···<im≤k

xi1xi2· · ·xim = X

α∈{0,1}^k;

|α|=m

x^α ∈ S.

(Thus, e0 = 1, and em= 0 when m<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

e_ν =e_ν1e_ν2· · ·e_ν` ∈ S.

Theorem (Gauss): The commutativek-algebraS is freely generated by the elementary symmetric polynomials

e1,e2, . . . ,ek. (That is, it is generated by them, and they are algebraically independent.)

Equivalent restatement: (eλ)_λis a partition whose entries are≤k

is a basis of the k-module S. Note thatem= 0 when m>k.

8 / 31

Symmetric polynomials: the e-basis

For eachm∈Z, we lete_m denote them-thelementary symmetric polynomial:

em= X

1≤i₁<i2<···<im≤k

xi1xi2· · ·xim = X

α∈{0,1}^k;

|α|=m

x^α ∈ S.

(Thus, e0 = 1, and em= 0 when m<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

e_ν =e_ν1e_ν2· · ·e_ν` ∈ S.

Theorem (Gauss): The commutativek-algebraS is freely generated by the elementary symmetric polynomials

e1,e2, . . . ,ek. (That is, it is generated by them, and they are algebraically independent.)

Equivalent restatement: (eλ)_λis a partition whose entries are≤k

is a basis of the k-moduleS.

Note thatem= 0 when m>k.

8 / 31

Symmetric polynomials: the e-basis

For eachm∈Z, we lete_m denote them-thelementary symmetric polynomial:

em= X

1≤i₁<i2<···<im≤k

xi1xi2· · ·xim = X

α∈{0,1}^k;

|α|=m

x^α ∈ S.

(Thus, e0 = 1, and em= 0 when m<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

e_ν =e_ν1e_ν2· · ·e_ν` ∈ S.

Theorem (Gauss): The commutativek-algebraS is freely generated by the elementary symmetric polynomials

e1,e2, . . . ,ek. (That is, it is generated by them, and they are algebraically independent.)

Equivalent restatement: (eλ)_λis a partition whose entries are≤k

is a basis of the k-moduleS. Note thatem= 0 when m>k.

8 / 31

Symmetric polynomials: the e-basis

For eachm∈Z, we lete_m denote them-thelementary symmetric polynomial:

em= X

1≤i₁<i2<···<im≤k

xi1xi2· · ·xim = X

α∈{0,1}^k;

|α|=m

x^α ∈ S.

(Thus, e0 = 1, and em= 0 when m<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

e_ν =e_ν1e_ν2· · ·e_ν` ∈ S.

Theorem (Gauss): The commutativek-algebraS is freely generated by the elementary symmetric polynomials

e1,e2, . . . ,ek. (That is, it is generated by them, and they are algebraically independent.)

Equivalent restatement: (eλ)_λis a partition whose entries are≤k

is a basis of the k-moduleS. Note thatem= 0 when m>k.

8 / 31

Symmetric polynomials: the h-bases

For eachm∈Z, we lethm denote them-thcomplete homogeneous symmetric polynomial:

hm = X

1≤i₁≤i₂≤···≤i_m≤k

x_i1x_i2· · ·x_im= X

α∈N^k;

|α|=m

x^α∈ S.

(Thus, h0 = 1, and hm = 0 whenm<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

h_ν =h_ν1h_ν2· · ·h_ν` ∈ S.

Theorem: The commutativek-algebra S is freely generated by the complete homogeneous symmetric polynomials h1,h2, . . . ,h_k.

Equivalent restatement: (hλ)_λis a partition whose entries are≤k

is a basis of the k-module S.

Theorem: (hλ)_{λis a}k-partition is a basis of the k-moduleS. (Another basis!)

9 / 31

Symmetric polynomials: the h-bases

For eachm∈Z, we lethm denote them-thcomplete homogeneous symmetric polynomial:

hm = X

1≤i₁≤i₂≤···≤i_m≤k

x_i1x_i2· · ·x_im= X

α∈N^k;

|α|=m

x^α∈ S.

(Thus, h0 = 1, and hm = 0 whenm<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

h_ν =h_ν1h_ν2· · ·h_ν` ∈ S.

Theorem: The commutativek-algebra S is freely generated by the complete homogeneous symmetric polynomials h1,h2, . . . ,h_k.

Equivalent restatement: (hλ)_λis a partition whose entries are≤k

is a basis of the k-module S.

Theorem: (hλ)_{λis a}k-partition is a basis of the k-moduleS. (Another basis!)

9 / 31

Symmetric polynomials: the h-bases

For eachm∈Z, we lethm denote them-thcomplete homogeneous symmetric polynomial:

hm = X

1≤i₁≤i₂≤···≤i_m≤k

x_i1x_i2· · ·x_im= X

α∈N^k;

|α|=m

x^α∈ S.

(Thus, h0 = 1, and hm = 0 whenm<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

h_ν =h_ν1h_ν2· · ·h_ν` ∈ S.

Theorem: The commutativek-algebra S is freely generated by the complete homogeneous symmetric polynomials h1,h2, . . . ,h_k.

Equivalent restatement: (hλ)_λis a partition whose entries are≤k

is a basis of the k-moduleS.

Theorem: (hλ)_{λis a}k-partition is a basis of the k-moduleS. (Another basis!)

9 / 31

Symmetric polynomials: the h-bases

For eachm∈Z, we lethm denote them-thcomplete homogeneous symmetric polynomial:

hm = X

1≤i₁≤i₂≤···≤i_m≤k

x_i1x_i2· · ·x_im= X

α∈N^k;

|α|=m

x^α∈ S.

(Thus, h0 = 1, and hm = 0 whenm<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

h_ν =h_ν1h_ν2· · ·h_ν` ∈ S.

Theorem: The commutativek-algebra S is freely generated by the complete homogeneous symmetric polynomials h1,h2, . . . ,h_k.

Equivalent restatement: (hλ)_λis a partition whose entries are≤k

is a basis of the k-moduleS.

Theorem: (hλ)_{λis a}k-partition is a basis of the k-moduleS.

(Another basis!)

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Symmetric polynomials: the h-bases

For eachm∈Z, we lethm denote them-thcomplete homogeneous symmetric polynomial:

hm = X

1≤i₁≤i₂≤···≤i_m≤k

x_i1x_i2· · ·x_im= X

α∈N^k;

|α|=m

x^α∈ S.

(Thus, h0 = 1, and hm = 0 whenm<0.)

For eachν = (ν1, ν2, . . . , ν`)∈Z<sup>`</sup> (e.g., ak-partition when

`=k), set

h_ν =h_ν1h_ν2· · ·h_ν` ∈ S.

Theorem: The commutativek-algebra S is freely generated by the complete homogeneous symmetric polynomials h1,h2, . . . ,h_k.

Equivalent restatement: (hλ)_λis a partition whose entries are≤k

is a basis of the k-moduleS.

Theorem: (hλ)_{λis a}k-partition is a basis of the k-moduleS.

(Another basis!)

9 / 31

Symmetric polynomials: the s-basis (Schur polynomials) For eachk-partitionλ, we lets_λ be theλ-th Schur polynomial:

s_λ= det

x_i^λj+k−j

1≤i≤k,1≤j≤k

det

x_i^k−j

1≤i≤k,1≤j≤k

(alternant formula)

= det

(hλ_i−i+j)_{1≤i≤k,1≤j≤k}

(Jacobi-Trudi). Theorem: The equality above holds, and sλ is a symmetric polynomial with nonnegative coefficients.

Theorem: (s_λ)_{λis a}k-partition is a basis of the k-moduleS.

10 / 31

Symmetric polynomials: the s-basis (Schur polynomials) For eachk-partitionλ, we lets_λ be theλ-th Schur polynomial:

s_λ= det

x_i^λj+k−j

1≤i≤k,1≤j≤k

det

x_i^k−j

1≤i≤k,1≤j≤k

(alternant formula)

= det

(hλ_i−i+j)_{1≤i≤k,1≤j≤k}

(Jacobi-Trudi). Theorem: The equality above holds, and sλ is a symmetric polynomial with nonnegative coefficients. Explicitly,

sλ = X

Tis a semistandardλ-tableau with entries 1,2,...,k

k

Y

i=1

x_i^{(number ofi’s inT)},

where a semistandardλ-tableau with entries1,2, . . . ,k is a way of putting an integer i ∈ {1,2, . . . ,k}into each box of Y (λ) such that the entriesweakly increase along rows and strictly increase along columns.

Theorem: (s_λ)_{λis ak-partition} is a basis of the k-module S. ^{10 / 31}

Symmetric polynomials: the s-basis (Schur polynomials) For eachk-partitionλ, we lets_λ be theλ-th Schur polynomial:

s_λ= det

x_i^λj+k−j

1≤i≤k,1≤j≤k

det

x_i^k−j

1≤i≤k,1≤j≤k

(alternant formula)

= det

(hλ_i−i+j)_{1≤i≤k,1≤j≤k}

(Jacobi-Trudi). Theorem: The equality above holds, and sλ is a symmetric polynomial with nonnegative coefficients.

Theorem: (s_λ)_{λis ak-partition} is a basis of the k-module S.

10 / 31

Symmetric polynomials: Littlewood-Richardson coefficients Ifλandµ are twok-partitions, then the product s_λs_µ can be again written as a k-linear combination of Schur polynomials (since these form a basis):

s_λs_µ= X

νis ak-partition

c_λ,µ^ν s_ν,

where the c_λ,µ^ν lie ink. Thesec_λ,µ^ν are called the Littlewood-Richardson coefficients.

Theorem: These Littlewood-Richardson coefficients c_λ,µ^ν are nonnegative integers (and count something).

11 / 31

Symmetric polynomials: Littlewood-Richardson coefficients Ifλandµ are twok-partitions, then the product s_λs_µ can be again written as a k-linear combination of Schur polynomials (since these form a basis):

s_λs_µ= X

νis ak-partition

c_λ,µ^ν s_ν,

where the c_λ,µ^ν lie ink. Thesec_λ,µ^ν are called the Littlewood-Richardson coefficients.

Theorem: These Littlewood-Richardson coefficients c_λ,µ^ν are nonnegative integers (and count something).

11 / 31

Symmetric polynomials: Schur polynomials for non-partitions We have defined

s_λ = det

(h_λi−i+j)_{1≤i≤k,1≤j≤k} for k-partitions λ.

Apply the same definition to arbitrary λ∈Z^k.

Proposition: Ifα∈Z^k, then sα is either 0 or equals ±s_λ for somek-partitionλ.

More precisely: Let

β = (α1+ (k−1), α2+ (k−2), . . . , αk+ (k−k)). Ifβ has a negative entry, thens_α= 0.

Ifβ has two equal entries, thens_α = 0.

Otherwise, letγ be the k-tuple obtained by sorting β in decreasing order, and letσ be the permutation of the indices that causes this sorting. Let λbe the k-partition (γ1−(k−1), γ2−(k−2), . . . , γk−(k−k)). Then, sα = (−1)^σs_λ.

Also, the alternant formula still holds if all λ_i + (k−i) are

≥0.

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Symmetric polynomials: Schur polynomials for non-partitions We have defined

s_λ = det

(h_λi−i+j)_{1≤i≤k,1≤j≤k} for k-partitions λ.

Apply the same definition to arbitrary λ∈Z^k.

Proposition: Ifα∈Z^k, then sα is either 0 or equals ±s_λ for somek-partitionλ.

(So we get nothing really new.) More precisely: Let

β = (α₁+ (k−1), α₂+ (k−2), . . . , α_k+ (k−k)).

Ifβ has a negative entry, thens_α= 0. Ifβ has two equal entries, thensα = 0.

Otherwise, letγ be the k-tuple obtained by sorting β in decreasing order, and letσ be the permutation of the indices that causes this sorting. Let λbe the k-partition (γ1−(k−1), γ2−(k−2), . . . , γ_k−(k−k)). Then, s_α = (−1)^σs_λ.

Also, the alternant formula still holds if all λi + (k−i) are

≥0.

12 / 31

Symmetric polynomials: Schur polynomials for non-partitions We have defined

s_λ = det

(h_λi−i+j)_{1≤i≤k,1≤j≤k} for k-partitions λ.

Apply the same definition to arbitrary λ∈Z^k.

Proposition: Ifα∈Z^k, then sα is either 0 or equals ±s_λ for somek-partitionλ.

More precisely: Let

β = (α1+ (k−1), α2+ (k−2), . . . , αk+ (k−k)).

Ifβ has a negative entry, thens_α= 0.

Ifβ has two equal entries, thens_α = 0.

Otherwise, letγ be the k-tuple obtained by sorting β in decreasing order, and letσ be the permutation of the indices that causes this sorting. Let λbe the k-partition (γ1−(k−1), γ2−(k−2), . . . , γk−(k−k)). Then, s_α = (−1)^σs_λ.

Also, the alternant formula still holds if all λ_i + (k−i) are

≥0.

12 / 31

Symmetric polynomials: Schur polynomials for non-partitions We have defined

s_λ = det

(h_λi−i+j)_{1≤i≤k,1≤j≤k} for k-partitions λ.

Apply the same definition to arbitrary λ∈Z^k.

Proposition: Ifα∈Z^k, then sα is either 0 or equals ±s_λ for somek-partitionλ.

More precisely: Let

β = (α1+ (k−1), α2+ (k−2), . . . , αk+ (k−k)).

Ifβ has a negative entry, thens_α= 0.

Ifβ has two equal entries, thens_α = 0.

Otherwise, letγ be the k-tuple obtained by sorting β in decreasing order, and letσ be the permutation of the indices that causes this sorting. Let λbe the k-partition (γ1−(k−1), γ2−(k−2), . . . , γk−(k−k)). Then, s_α = (−1)^σs_λ.

Also, the alternant formula still holds if all λ_i + (k−i) are

≥0. _{12 / 31}

Symmetric polynomials: Schur polynomials for non-partitions We have defined

s_λ = det

(h_λi−i+j)_{1≤i≤k,1≤j≤k} for k-partitions λ.

Apply the same definition to arbitrary λ∈Z^k.

Proposition: Ifα∈Z^k, then sα is either 0 or equals ±s_λ for somek-partitionλ.

More precisely: Let

β = (α1+ (k−1), α2+ (k−2), . . . , αk+ (k−k)).

Ifβ has a negative entry, thens_α= 0.

Ifβ has two equal entries, thens_α = 0.

Otherwise, letγ be the k-tuple obtained by sorting β in decreasing order, and letσ be the permutation of the indices that causes this sorting. Let λbe the k-partition (γ1−(k−1), γ2−(k−2), . . . , γk−(k−k)). Then, s_α = (−1)^σs_λ.

Also, the alternant formula still holds if all λ_i + (k−i) are

≥0. _{12 / 31}

A more general setting: a1,a2, . . . ,a_k and J

Pick any integer n≥k.

Let a1,a2, . . . ,ak ∈ P such that degai <n−k+i for all i. (For example, this holds if a_i ∈k.)

Let J be the ideal of P generated by the k differences h_n−k+1−a₁, h_n−k+2−a₂, . . . , h_n−a_k.

Theorem (G.): The k-module PJ is free with basis (x^α)_α∈

N^k;αi<n−k+ifor eachi

informally: “

x₁^<n−k+1x₂^<n−k+2· · ·x_n^<n

”

where the overline means “projection” onto whatever quotient we need (here: from P ontoPJ).

(This basis has n(n−1)· · ·(n−k+ 1) elements.)

13 / 31

A more general setting: a1,a2, . . . ,a_k and J

Pick any integer n≥k.

Let a1,a2, . . . ,ak ∈ P such that degai <n−k+i for all i. (For example, this holds if a_i ∈k.)

Let J be the ideal ofP generated by thek differences h_n−k+1−a₁, h_n−k+2−a₂, . . . , h_n−a_k.

Theorem (G.): The k-module PJ is free with basis (x^α)_α∈

N^k;αi<n−k+ifor eachi

informally: “

x₁^<n−k+1x₂^<n−k+2· · ·x_n^<n

”

where the overline means “projection” onto whatever quotient we need (here: from P ontoPJ).

(This basis has n(n−1)· · ·(n−k+ 1) elements.)

13 / 31

A more general setting: a1,a2, . . . ,a_k and J

Pick any integer n≥k.

Let a1,a2, . . . ,ak ∈ P such that degai <n−k+i for all i. (For example, this holds if a_i ∈k.)

Let J be the ideal ofP generated by thek differences h_n−k+1−a₁, h_n−k+2−a₂, . . . , h_n−a_k.

Theorem (G.): The k-module PJ is free with basis (x^α)_α∈

N^k;αi<n−k+ifor eachi

informally: “

x₁^<n−k+1x₂^<n−k+2· · ·x_n^<n

”

where the overline means “projection” onto whatever quotient we need (here: from P ontoPJ).

(This basis has n(n−1)· · ·(n−k+ 1) elements.)

13 / 31

A more general setting: a1,a2, . . . ,a_k and J

Pick any integer n≥k.

Let a1,a2, . . . ,ak ∈ P such that degai <n−k+i for all i. (For example, this holds if a_i ∈k.)

Let J be the ideal ofP generated by thek differences h_n−k+1−a₁, h_n−k+2−a₂, . . . , h_n−a_k.

Theorem (G.): The k-module PJ is free with basis (x^α)_α∈

N^k;αi<n−k+ifor eachi

informally: “

x₁^<n−k+1x₂^<n−k+2· · ·x_n^<n

”

where the overline means “projection” onto whatever quotient we need (here: from P ontoPJ).

(This basis has n(n−1)· · ·(n−k+ 1) elements.)

13 / 31

A slightly less general setting: symmetric a1,a2, . . . ,a_k and J

FROM NOW ON, assume that a₁,a₂, . . . ,a_k ∈ S.

Let I be the ideal ofS generated by the k differences hn−k+1−a₁, hn−k+2−a₂, . . . , h_n−a_k. (Same differences as forJ, but we are generating an ideal of S now.)

Let ω= (n−k,n−k, . . . ,n−k)

| {z }

kentries

and

Pk,n ={λis a k-partition | λ1≤n−k}

={k-partitions λ⊆ω}.

Here, for two k-partitions α andβ, we say that α⊆β if and only if αi ≤βi for all i.

Theorem (G.): The k-module SI is free with basis (sλ)_λ∈P

k,n.

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A slightly less general setting: symmetric a1,a2, . . . ,a_k and J

FROM NOW ON, assume that a₁,a₂, . . . ,a_k ∈ S.

Let I be the ideal ofS generated by the k differences hn−k+1−a₁, hn−k+2−a₂, . . . , h_n−a_k.

(Same differences as forJ, but we are generating an ideal of S now.)

Let ω= (n−k,n−k, . . . ,n−k)

| {z }

kentries

and

Pk,n={λis a k-partition | λ1≤n−k}

={k-partitions λ⊆ω}.

Here, for two k-partitions α andβ, we say that α⊆β if and only if αi ≤βi for all i.

Theorem (G.): The k-module SI is free with basis (sλ)_λ∈P

k,n.

14 / 31

A slightly less general setting: symmetric a1,a2, . . . ,a_k and J

FROM NOW ON, assume that a₁,a₂, . . . ,a_k ∈ S.

Let I be the ideal ofS generated by the k differences hn−k+1−a₁, hn−k+2−a₂, . . . , h_n−a_k.

(Same differences as forJ, but we are generating an ideal of S now.)

Let ω= (n−k,n−k, . . . ,n−k)

| {z }

kentries

and

Pk,n={λis a k-partition | λ1≤n−k}

={k-partitions λ⊆ω}.

Here, for two k-partitions α andβ, we say that α⊆β if and only if αi ≤βi for all i.

Theorem (G.): The k-module SI is free with basis (sλ)_λ∈P

k,n.

14 / 31

An even less general setting: constant a1,a2, . . . ,a_k

FROM NOW ON, assume that a1,a2, . . . ,ak ∈k.

This setting still is general enough to encompass ...

classical cohomology: Ifk=Zand

a₁=a₂=· · ·=a_k = 0, thenSI becomes the cohomology ring H^∗(Gr (k,n)); the basis (s_λ)_λ∈P

k,n

corresponds to the Schubert classes.

quantum cohomology: Ifk=Z[q] and

a₁=a₂=· · ·=a_k−1 = 0 anda_k =−(−1)^kq, then SI becomes the quantum cohomology ring

QH^∗(Gr (k,n)).

The above theorem lets us work in these rings (and more generally) without relying on geometry.

15 / 31

An even less general setting: constant a1,a2, . . . ,a_k

FROM NOW ON, assume that a1,a2, . . . ,ak ∈k.

This setting still is general enough to encompass ...

classical cohomology: Ifk=Zand

a₁=a₂=· · ·=a_k = 0, thenSI becomes the cohomology ring H^∗(Gr (k,n)); the basis (s_λ)_λ∈P

k,n

corresponds to the Schubert classes.

quantum cohomology: Ifk=Z[q] and

a₁=a₂=· · ·=a_k−1 = 0 anda_k =−(−1)^kq, then SI becomes the quantum cohomology ring

QH^∗(Gr (k,n)).

The above theorem lets us work in these rings (and more generally) without relying on geometry.

15 / 31

An even less general setting: constant a1,a2, . . . ,a_k

FROM NOW ON, assume that a1,a2, . . . ,ak ∈k.

This setting still is general enough to encompass ...

classical cohomology: Ifk=Zand

a₁=a₂=· · ·=a_k = 0, thenSI becomes the cohomology ring H^∗(Gr (k,n)); the basis (s_λ)_λ∈P

k,n

corresponds to the Schubert classes.

quantum cohomology: Ifk=Z[q] and

a₁=a₂=· · ·=a_k−1 = 0 anda_k =−(−1)^kq, then SI becomes the quantum cohomology ring

QH^∗(Gr (k,n)).

The above theorem lets us work in these rings (and more generally) without relying on geometry.

15 / 31

S3-symmetry of the Gromov–Witten invariants Recall that (s_λ)_λ∈P

k,n is a basis of the k-moduleSI.

For each µ∈Pk,n, let coeffµ:SI →ksend each element to its s_µ-coordinate wrt this basis.

For every k-partition ν= (ν₁, ν₂, . . . , ν_k)∈P_k,n, we define ν^∨:= (n−k−ν_k,n−k−νk−1, . . . ,n−k−ν₁)∈P_k,n. This k-partition ν^∨ is called the complementof ν.

For any three k-partitionsα, β, γ ∈P_k,n, let g_α,β,γ:= coeff_γ^∨(s_αs_β)∈k.

These generalize the Littlewood–Richardson coefficients and (3-point) Gromov–Witten invariants.

Theorem (G.): For anyα, β, γ∈P_k,n, we have

g_α,β,γ =g_α,γ,β=g_β,α,γ =g_β,γ,α=g_γ,α,β=g_γ,β,α

= coeff_ω(s_αs_βs_γ).

Equivalent restatement: Each ν∈P_k,n andf ∈ SI satisfy coeff_ω(s_νf) = coeff_ν^∨(f).

16 / 31

S3-symmetry of the Gromov–Witten invariants Recall that (s_λ)_λ∈P

k,n is a basis of the k-moduleSI.

For each µ∈Pk,n, let coeffµ:SI →ksend each element to its s_µ-coordinate wrt this basis.

For every k-partition ν= (ν₁, ν₂, . . . , ν_k)∈P_k,n, we define ν^∨:= (n−k−ν_k,n−k−νk−1, . . . ,n−k−ν₁)∈P_k,n. Thisk-partitionν^∨ is called the complementof ν.

For any three k-partitionsα, β, γ ∈P_k,n, let g_α,β,γ:= coeff_γ^∨(s_αs_β)∈k.

These generalize the Littlewood–Richardson coefficients and (3-point) Gromov–Witten invariants.

Theorem (G.): For anyα, β, γ∈P_k,n, we have

g_α,β,γ =g_α,γ,β=g_β,α,γ =g_β,γ,α=g_γ,α,β=g_γ,β,α

= coeff_ω(s_αs_βs_γ).

Equivalent restatement: Each ν∈P_k,n andf ∈ SI satisfy coeff_ω(s_νf) = coeff_ν^∨(f).

16 / 31