Massachusetts Institute of Technology

A quotient of the ring of symmetric functions generalizing quantum cohomology

Darij Grinberg

7 December 2018

Massachusetts Institute of Technology, Cambridge, MA slides: http:

//www.cip.ifi.lmu.de/~grinberg/algebra/mit2018.pdf paper: http:

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

What is this about?

From a modern point of view, Schubert calculus 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).

What is this about?

From a modern point of view, Schubert calculus 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).

What is this about?

From a modern point of view, Schubert calculus 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).

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, with λranging over all partitions with ≤k parts and each part≤n−k. The structure constants are theGromov–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.

Quantum cohomology of Gr(k,n)

For comparison, the classical cohomology of the Grassmannian is

H^∗(Gr (k,n))

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

(where the hi are complete homogeneous symmetric polynomials).

Many properties of classical cohomology still hold here. In particular: QH^∗(Gr (k,n)) has a Z[q]-module basis (s_λ)_λ∈P

k,n

of (projected) Schur polynomials, with λranging over all partitions with ≤k parts and each part≤n−k. The structure constants are theGromov–Witten invariants.

References:

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

Alexander Postnikov, Affine approach to quantum

3 / 24

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, with λranging over all partitions with ≤k parts and each part≤n−k. The structure constants are theGromov–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.

A more general setting: P and S

We will now deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

Let kbe a commutative ring. LetN={0,1,2, . . .}. Let n ≥k ≥0 be integers.

Let P =k[x₁,x₂, . . . ,x_k].

For eachα∈N^k and each i ∈ {1,2, . . . ,k}, letαi be the i-th entry of α. Same for infinite sequences (like partitions). 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. Theorem (Artin ≤1944): TheS-moduleP is free with basis

(x^α)_α∈Nk;αi<i for eachi.

A more general setting: P and S

We will now deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

Let kbe a commutative ring. LetN={0,1,2, . . .}. Let n ≥k ≥0 be integers.

Let P =k[x₁,x₂, . . . ,x_k].

For eachα∈N^k and each i ∈ {1,2, . . . ,k}, letαi be the i-th entry of α. Same for infinite sequences (like partitions). 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. Theorem (Artin ≤1944): TheS-moduleP is free with basis

(x^α)_α∈Nk;αi<i for eachi.

A more general setting: P and S

We will now deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

Let kbe a commutative ring. LetN={0,1,2, . . .}. Let n ≥k ≥0 be integers.

Let P =k[x₁,x₂, . . . ,x_k].

For eachα∈N^k and eachi ∈ {1,2, . . . ,k}, letα_i be the i-th entry of α. Same for infinite sequences (like partitions).

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. Theorem (Artin ≤1944): TheS-moduleP is free with basis

(x^α)_α∈Nk;αi<i for eachi.

A more general setting: P and S

We will now deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

Let kbe a commutative ring. LetN={0,1,2, . . .}. Let n ≥k ≥0 be integers.

Let P =k[x₁,x₂, . . . ,x_k].

For eachα∈N^k and eachi ∈ {1,2, . . . ,k}, letα_i be the i-th entry of α. Same for infinite sequences (like partitions).

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. Theorem (Artin ≤1944): TheS-moduleP is free with basis

(x^α)_α∈Nk;αi<i for eachi.

A more general setting: P and S

We will now deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

Let kbe a commutative ring. LetN={0,1,2, . . .}. Let n ≥k ≥0 be integers.

Let P =k[x₁,x₂, . . . ,x_k].

For eachα∈N^k and eachi ∈ {1,2, . . . ,k}, letα_i be the i-th entry of α. Same for infinite sequences (like partitions).

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.

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

A more general setting: P and S

We will now deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

Let kbe a commutative ring. LetN={0,1,2, . . .}. Let n ≥k ≥0 be integers.

Let P =k[x₁,x₂, . . . ,x_k].

For eachα∈N^k and eachi ∈ {1,2, . . . ,k}, letα_i be the i-th entry of α. Same for infinite sequences (like partitions).

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. Theorem (Artin ≤1944): TheS-moduleP is free with basis

(x^α)_α∈Nk;αi<i for eachi.

A more general setting: P and S

We will now deform H^∗(Gr (k,n)) using k parameters instead of one, generalizing QH^∗(Gr (k,n)).

Let kbe a commutative ring. LetN={0,1,2, . . .}. Let n ≥k ≥0 be integers.

Let P =k[x₁,x₂, . . . ,x_k].

For eachα∈N^k and eachi ∈ {1,2, . . . ,k}, letα_i be the i-th entry of α. Same for infinite sequences (like partitions).

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. Theorem (Artin ≤1944): TheS-moduleP is free with basis

(x^α)_α∈Nk;αi<i for eachi.

Reminders on symmetric polynomials

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

We need the following ones:

Reminders on symmetric polynomials: the e-basis

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

e_m= X

1≤i1<i2<···<im≤k

x_i1x_i2· · ·x_im = X

α∈{0,1}^k;

|α|=m

x^α ∈ S.

(Thus, e₀ = 1, and e_m= 0 when m<0.)

For eachν = (ν1, ν2, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set eν =eν1eν2· · ·eν` ∈ S.

Then, (e_λ)_λis a partition withλ1≤k is a basis of the k-moduleS. (Gauss’s theorem.)

Note thatem= 0 when m>k.

Reminders on symmetric polynomials: the e-basis

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

e_m= X

1≤i1<i2<···<im≤k

x_i1x_i2· · ·x_im = X

α∈{0,1}^k;

|α|=m

x^α ∈ S.

(Thus, e₀ = 1, and e_m= 0 when m<0.)

For eachν = (ν1, ν2, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set eν =eν1eν2· · ·eν` ∈ S.

Then, (e_λ)_λis a partition withλ1≤k is a basis of the k-moduleS. (Gauss’s theorem.)

Note thatem= 0 when m>k.

Reminders on symmetric polynomials: the e-basis

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

e_m= X

1≤i1<i2<···<im≤k

x_i1x_i2· · ·x_im = X

α∈{0,1}^k;

|α|=m

x^α ∈ S.

(Thus, e₀ = 1, and e_m= 0 when m<0.)

For eachν = (ν1, ν2, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set eν =eν1eν2· · ·eν` ∈ S.

Then, (e_λ)_λis a partition withλ1≤k is a basis of the k-moduleS. (Gauss’s theorem.)

Note thatem= 0 when m>k.

Reminders on symmetric polynomials: the e-basis

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

e_m= X

1≤i1<i2<···<im≤k

x_i1x_i2· · ·x_im = X

α∈{0,1}^k;

|α|=m

x^α ∈ S.

(Thus, e₀ = 1, and e_m= 0 when m<0.)

For eachν = (ν1, ν2, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set eν =eν1eν2· · ·eν` ∈ S.

Then, (e_λ)_λis a partition withλ1≤k is a basis of the k-moduleS. (Gauss’s theorem.)

Note thatem= 0 when m>k.

Reminders on symmetric polynomials: the h-bases

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

h_m = X

1≤i1≤i2≤···≤im≤k

x_i1x_i2· · ·x_im= X

α∈N^k;

|α|=m

x^α∈ S.

(Thus, h₀ = 1, and h_m = 0 whenm<0.)

For eachν = (ν₁, ν₂, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set h_ν =h_ν1h_ν2· · ·h_ν` ∈ S.

Then, (hλ)_λis a partition withλ1≤k is a basis of the k-module S. Also, (hλ)_λis a partition with`(λ)≤k is a basis of the k-module S. Here, `(λ) is the length of λ, that is, the number of parts (= nonzero entries) of λ.

Reminders on symmetric polynomials: the h-bases

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

h_m = X

1≤i1≤i2≤···≤im≤k

x_i1x_i2· · ·x_im= X

α∈N^k;

|α|=m

x^α∈ S.

(Thus, h₀ = 1, and h_m = 0 whenm<0.)

For eachν = (ν₁, ν₂, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set h_ν =h_ν1h_ν2· · ·h_ν` ∈ S.

Then, (h_λ)_λis a partition withλ1≤k is a basis of the k-module S.

Also, (hλ)_λis a partition with`(λ)≤k is a basis of the k-module S. Here, `(λ) is the length of λ, that is, the number of parts (= nonzero entries) of λ.

Reminders on symmetric polynomials: the h-bases

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

h_m = X

1≤i1≤i2≤···≤im≤k

x_i1x_i2· · ·x_im= X

α∈N^k;

|α|=m

x^α∈ S.

(Thus, h₀ = 1, and h_m = 0 whenm<0.)

For eachν = (ν₁, ν₂, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set h_ν =h_ν1h_ν2· · ·h_ν` ∈ S.

Then, (h_λ)_λis a partition withλ1≤k is a basis of the k-module S.

Also, (hλ)_λis a partition with`(λ)≤k is a basis of the k-module S.

Here, `(λ) is the length of λ, that is, the number of parts (=

nonzero entries) of λ.

Reminders on symmetric polynomials: the h-bases

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

h_m = X

1≤i1≤i2≤···≤im≤k

x_i1x_i2· · ·x_im= X

α∈N^k;

|α|=m

x^α∈ S.

(Thus, h₀ = 1, and h_m = 0 whenm<0.)

For eachν = (ν₁, ν₂, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set h_ν =h_ν1h_ν2· · ·h_ν` ∈ S.

Then, (h_λ)_λis a partition withλ1≤k is a basis of the k-module S.

Also, (hλ)_λis a partition with`(λ)≤k is a basis of the k-module S.

Here, `(λ) is the length of λ, that is, the number of parts (=

nonzero entries) of λ.

Reminders on symmetric polynomials: the s-basis

For each partitionλ= (λ₁, λ₂, λ₃, . . .), we lets_λ be the λ-th Schur polynomial:

s_λ= X

T is a semistandard tableau of shapeλwith entries 1,2,...,k

k

Y

i=1

x_i^{(number ofi’s inT)}

= det

(h_λi−i+j)_{1≤i≤`(λ),1≤j≤`(λ)}

(Jacobi-Trudi). If`(λ)>k, then s_λ = 0.

If`(λ)≤k, then

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).

Now, (s_λ)_λis a partition with`(λ)≤k is a basis of the k-moduleS.

Reminders on symmetric polynomials: the s-basis

For each partitionλ= (λ₁, λ₂, λ₃, . . .), we lets_λ be the λ-th Schur polynomial:

s_λ= X

T is a semistandard tableau of shapeλwith entries 1,2,...,k

k

Y

i=1

x_i^{(number ofi’s inT)}

= det

(h_λi−i+j)_{1≤i≤`(λ),1≤j≤`(λ)}

(Jacobi-Trudi). If`(λ)>k, then s_λ = 0.

If`(λ)≤k, then

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).

Now, (s_λ)_λis a partition with`(λ)≤k is a basis of the k-module S.

Reminders on symmetric polynomials: the s-basis

For each partitionλ= (λ₁, λ₂, λ₃, . . .), we lets_λ be the λ-th Schur polynomial:

s_λ= X

T is a semistandard tableau of shapeλwith entries 1,2,...,k

k

Y

i=1

x_i^{(number ofi’s inT)}

= det

(h_λi−i+j)_{1≤i≤`(λ),1≤j≤`(λ)}

(Jacobi-Trudi). If`(λ)>k, then s_λ = 0.

If`(λ)≤k, then

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).

Now, (s_λ)_λis a partition with`(λ)≤k is a basis of the k-module S.

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

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

Let J be the ideal ofP generated by thek differences hn−k+1−a₁, hn−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+i for eachi,

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

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

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

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

Let J be the ideal ofP generated by thek differences hn−k+1−a₁, hn−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+i for eachi,

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

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

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

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

Let J be the ideal ofP generated by thek differences hn−k+1−a₁, hn−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+i for eachi,

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

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

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.)

For each partitionλ, let s_λ∈ S be the corresponding Schur polynomial.

Let

Pk,n={λis a partition | λ1≤n−k and `(λ)≤k}

={partitionsλ⊆ω}, where ω= (n−k,n−k, . . . ,n−k)

| {z }

k entries

.

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

k,n.

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.)

For each partitionλ, lets_λ∈ S be the corresponding Schur polynomial.

Let

Pk,n={λis a partition | λ1≤n−k and `(λ)≤k}

={partitionsλ⊆ω}, where ω= (n−k,n−k, . . . ,n−k)

| {z }

k entries

.

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

k,n.

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.)

For each partitionλ, lets_λ∈ S be the corresponding Schur polynomial.

Let

Pk,n ={λis a partition | λ1≤n−k and`(λ)≤k}

={partitionsλ⊆ω}, whereω = (n−k,n−k, . . . ,n−k)

| {z }

k entries

.

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

k,n.

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.)

For each partitionλ, lets_λ∈ S be the corresponding Schur polynomial.

Let

Pk,n ={λis a partition | λ1≤n−k and`(λ)≤k}

={partitionsλ⊆ω}, whereω = (n−k,n−k, . . . ,n−k)

| {z }

k entries

.

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

k,n.

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.)

For each partitionλ, lets_λ∈ S be the corresponding Schur polynomial.

Let

Pk,n ={λis a partition | λ1≤n−k and`(λ)≤k}

={partitionsλ⊆ω}, whereω = (n−k,n−k, . . . ,n−k)

| {z }

k entries

.

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

k,n.

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 several that we know:

Ifk=Z anda1=a2=· · ·=ak = 0, thenSI becomes the cohomology ring H^∗(Gr (k,n)); the basis (s_λ)_λ∈P

k,n

corresponds to the Schubert classes.

Ifk=Z[q] anda1 =a2 =· · ·=ak−1= 0 and a_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.

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 several that we know:

Ifk=Z anda1=a2=· · ·=ak = 0, thenSI becomes the cohomology ring H^∗(Gr (k,n)); the basis (s_λ)_λ∈P

k,n

corresponds to the Schubert classes.

Ifk=Z[q] anda1 =a2 =· · ·=ak−1= 0 and a_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.

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 several that we know:

Ifk=Z anda1=a2=· · ·=ak = 0, thenSI becomes the cohomology ring H^∗(Gr (k,n)); the basis (s_λ)_λ∈P

k,n

corresponds to the Schubert classes.

Ifk=Z[q] anda1 =a2 =· · ·=ak−1= 0 and a_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.

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 sµ-coordinate wrt this basis.

For every 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 partition ν^∨ is called the complement ofν.

For any three partitions α, β, γ∈P_k,n, let g_α,β,γ:= coeff_γ^∨(sαs_β)∈k.

These generalize the Littlewood–Richardson numbers 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).

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 sµ-coordinate wrt this basis.

For every 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 partition ν^∨ is called the complement ofν.

For any three partitions α, β, γ∈P_k,n, let g_α,β,γ:= coeff_γ^∨(sαs_β)∈k.

These generalize the Littlewood–Richardson numbers 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).

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 sµ-coordinate wrt this basis.

For every 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 partition ν^∨ is called the complement ofν.

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

These generalize the Littlewood–Richardson numbers 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).

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 sµ-coordinate wrt this basis.

For every 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 partition ν^∨ is called the complement ofν.

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

These generalize the Littlewood–Richardson numbers 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).

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 sµ-coordinate wrt this basis.

For every 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 partition ν^∨ is called the complement ofν.

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

These generalize the Littlewood–Richardson numbers 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).

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 sµ-coordinate wrt this basis.

For every 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 partition ν^∨ is called the complement ofν.

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

These generalize the Littlewood–Richardson numbers 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).

Theh-basis

Theorem (G.): The k-module SI is free with basis h_λ

λ∈P_k,n.

The transfer matrix between the two bases (sλ)_λ∈P

k,n and h_λ

λ∈P_k,n is unitriangular wrt the “size-then-anti-dominance”

order, but seems hard to describe.

Proposition (G.): Letm be a positive integer. Then, hn+m=

k−1

X

j=0

(−1)^ja_k−js_(m,1^j₎,

where m,1^j

:= (m,1,1, . . . ,1

| {z }

j ones

) (a hook-shaped partition).

Theh-basis

Theorem (G.): The k-module SI is free with basis h_λ

λ∈P_k,n.

The transfer matrix between the two bases (sλ)_λ∈P

k,n and h_λ

λ∈P_k,n is unitriangular wrt the “size-then-anti-dominance”

order, but seems hard to describe.

Proposition (G.): Letm be a positive integer. Then,

hn+m=

k−1

X

j=0

(−1)^ja_k−js_(m,1^j₎,

where m,1^j

:= (m,1,1, . . . ,1

| {z }

jones

) (a hook-shaped partition).

Theh-basis

Theorem (G.): The k-module SI is free with basis h_λ

λ∈P_k,n.

The transfer matrix between the two bases (sλ)_λ∈P

k,n and h_λ

λ∈P_k,n is unitriangular wrt the “size-then-anti-dominance”

order, but seems hard to describe.

Proposition (G.): Letm be a positive integer. Then, hn+m=

k−1

X

j=0

(−1)^ja_k−js_(m,1^j₎,

where m,1^j

:= (m,1,1, . . . ,1

| {z }

jones

) (a hook-shaped partition).

A Pieri rule

Theorem (G.): Let λ∈P_k,n. Letj ∈ {0,1, . . . ,n−k}.

Then,

s_λh_j = X

µ∈P_k,n; µλis a horizontalj-strip

s_µ−

k

X

i=1

(−1)ⁱa_iX

ν⊆λ

c_(n−k−j^λ _+1,1i−1),νs_ν,

wherec_α,β^γ are the usual Littlewood–Richardson coefficients.

This generalizes the Bertram/Ciocan-Fontanine/Fulton Pieri rule, but note that c_(n−k^λ _−j+1,1i−1),ν may be>1.

Example:

s_(4,3,2)h2 =s_(4,4,3)+a1 s_(4,2)+s_(3,2,1)+s_(3,3)

−a₂ s_(4,1)+s_(2,2,1)+s_(3,1,1)+ 2s_(3,2) +a3 s_(2,2)+s_(2,1,1)+s_(3,1)

. Multiplying by e_j appears harder:

s_(2,2,1)e₂=a₁s_(2,2)−2a₂s_(2,1)+a₃ s₍₂₎+s_(1,1)

+a²₁s₍₁₎−2a₁a₂s₍₎.

A Pieri rule

Theorem (G.): Let λ∈P_k,n. Letj ∈ {0,1, . . . ,n−k}.

Then,

s_λh_j = X

µ∈P_k,n; µλis a horizontalj-strip

s_µ−

k

X

i=1

(−1)ⁱa_iX

ν⊆λ

c_(n−k−j^λ _+1,1i−1),νs_ν,

wherec_α,β^γ are the usual Littlewood–Richardson coefficients.

This generalizes the Bertram/Ciocan-Fontanine/Fulton Pieri rule, but note that c_(n−k^λ _−j+1,1i−1),ν may be>1.

Example:

s_(4,3,2)h2=s_(4,4,3)+a1 s_(4,2)+s_(3,2,1)+s_(3,3)

−a₂ s_(4,1)+s_(2,2,1)+s_(3,1,1)+ 2s_(3,2) +a3 s_(2,2)+s_(2,1,1)+s_(3,1)

.

Multiplying by e_j appears harder:

s_(2,2,1)e₂=a₁s_(2,2)−2a₂s_(2,1)+a₃ s₍₂₎+s_(1,1)

+a²₁s₍₁₎−2a₁a₂s₍₎.

A Pieri rule

Theorem (G.): Let λ∈P_k,n. Letj ∈ {0,1, . . . ,n−k}.

Then,

s_λh_j = X

µ∈P_k,n; µλis a horizontalj-strip

s_µ−

k

X

i=1

(−1)ⁱa_iX

ν⊆λ

c_(n−k−j^λ _+1,1i−1),νs_ν,

wherec_α,β^γ are the usual Littlewood–Richardson coefficients.

This generalizes the Bertram/Ciocan-Fontanine/Fulton Pieri rule, but note that c_(n−k^λ _−j+1,1i−1),ν may be>1.

Example:

s_(4,3,2)h2=s_(4,4,3)+a1 s_(4,2)+s_(3,2,1)+s_(3,3)

−a₂ s_(4,1)+s_(2,2,1)+s_(3,1,1)+ 2s_(3,2) +a3 s_(2,2)+s_(2,1,1)+s_(3,1)

. Multiplying by ej appears harder:

s_(2,2,1)e₂=a₁s_(2,2)−2a₂s_(2,1)+a₃ s₍₂₎+s_(1,1)

+a²₁s₍₁₎−2a₁a₂s₍₎.

A Pieri rule

Theorem (G.): Let λ∈P_k,n. Letj ∈ {0,1, . . . ,n−k}.

Then,

s_λh_j = X

µ∈P_k,n; µλis a horizontalj-strip

s_µ−

k

X

i=1

(−1)ⁱa_iX

ν⊆λ

c_(n−k−j^λ _+1,1i−1),νs_ν,

wherec_α,β^γ are the usual Littlewood–Richardson coefficients.

This generalizes the Bertram/Ciocan-Fontanine/Fulton Pieri rule, but note that c_(n−k^λ _−j+1,1i−1),ν may be>1.

Example:

s_(4,3,2)h2=s_(4,4,3)+a1 s_(4,2)+s_(3,2,1)+s_(3,3)

−a₂ s_(4,1)+s_(2,2,1)+s_(3,1,1)+ 2s_(3,2) +a3 s_(2,2)+s_(2,1,1)+s_(3,1)

. Multiplying by ej appears harder:

s_(2,2,1)e₂=a₁s_(2,2)−2a₂s_(2,1)+a₃ s₍₂₎+s_(1,1)

+a²₁s₍₁₎−2a₁a₂s₍₎.

Positivity?

Conjecture: Let b_i = (−1)^n−k−1a_i for each i ∈ {1,2, . . . ,k}.

Let λ, µ, ν∈P_k,n. Then, (−1)|λ|+|µ|−|ν|

coeff_ν(s_λs_µ) is a polynomial inb1,b2, . . . ,bk with coefficients in N.

Verified for alln ≤7 using SageMath.

This would generalize positivity of Gromov–Witten invariants.

More questions

Question: DoesSI have a geometric meaning? If not, why does it behave so nicely?

Question: What other bases does SI have? Monomial symmetric? Power-sum?

Question: Do other properties of QH^∗(Gr (k,n)) (such as

“curious duality” and “cyclic hidden symmetry”) generalize to SI?

(The Gr(k,n)→Gr(n−k,n) duality isomorphism does not exist in general: Ifk=Canda1= 6 anda2= 16, then

(SI)_k=2,n=3,a

1=6,a₂=16∼=C[x]/ (x−10) (x+ 2)²

, which can never be a (SI)_k=1,n=3, since (SI)_{k=1, n=3}∼=C[x]/ x³−a1

.)

Question: Is there an analogous generalization of

QH^∗(Fl (n)) ? Is it connected to Fulton’s “universal Schubert polynomials”?

Question: Is there an equivariant analogue?

Question: “Straightening rules” for s_λ whenλ /∈P_k,n, similar to the Bertram/Ciocan-Fontanine/Fulton “rim hook

algorithm”?

More questions

Question: DoesSI have a geometric meaning? If not, why does it behave so nicely?

Question: What other bases does SI have? Monomial symmetric? Power-sum?

Question: Do other properties of QH^∗(Gr (k,n)) (such as

“curious duality” and “cyclic hidden symmetry”) generalize to SI?

(The Gr(k,n)→Gr(n−k,n) duality isomorphism does not exist in general: Ifk=Canda1= 6 anda2= 16, then

(SI)_k=2,n=3,a

1=6,a₂=16∼=C[x]/ (x−10) (x+ 2)²

, which can never be a (SI)_k=1,n=3, since (SI)_k=1,n=3∼=C[x]/ x³−a1

.)

Question: Is there an analogous generalization of

QH^∗(Fl (n)) ? Is it connected to Fulton’s “universal Schubert polynomials”?

Question: Is there an equivariant analogue?

Question: “Straightening rules” for s_λ whenλ /∈P_k,n, similar to the Bertram/Ciocan-Fontanine/Fulton “rim hook

algorithm”?

More questions

Question: DoesSI have a geometric meaning? If not, why does it behave so nicely?

Question: What other bases does SI have? Monomial symmetric? Power-sum?

Question: Do other properties of QH^∗(Gr (k,n)) (such as

“curious duality” and “cyclic hidden symmetry”) generalize to SI?

(The Gr(k,n)→Gr(n−k,n) duality isomorphism does not exist in general: Ifk=Canda1= 6 anda2= 16, then

(SI)_k=2,n=3,a

1=6,a₂=16∼=C[x]/ (x−10) (x+ 2)²

, which can never be a (SI)_k=1,n=3, since (SI)_k=1,n=3∼=C[x]/ x³−a1

.)

Question: Is there an analogous generalization of

QH^∗(Fl (n)) ? Is it connected to Fulton’s “universal Schubert polynomials”?

Question: Is there an equivariant analogue?

Question: “Straightening rules” for s_λ whenλ /∈P_k,n, similar to the Bertram/Ciocan-Fontanine/Fulton “rim hook

algorithm”?