University of Connecticut

A quotient of the ring of symmetric functions generalizing quantum cohomology

Darij Grinberg

5 December 2018

University of Connecticut, Storrs slides: http:

//www.cip.ifi.lmu.de/~grinberg/algebra/uconn2018.pdf paper: http:

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

1 / 20

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).

2 / 20

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).

2 / 20

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).

2 / 20

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 from classical cohomology still hold. 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.

3 / 20

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 from classical cohomology still hold. 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.

3 / 20

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.

4 / 20

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.

4 / 20

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.

4 / 20

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.

4 / 20

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.

4 / 20

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.

4 / 20

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.

4 / 20

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 a_i ∈k.)

For eachm∈Z, we leth_m 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 hm = 0 whenm<0.)

Let J be the ideal of P generated by the k differences hn−k+1−a1, hn−k+2−a2, . . . , hn−ak. 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.)

5 / 20

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 a_i ∈k.)

For eachm∈Z, we leth_m 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 hm = 0 whenm<0.)

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

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.)

5 / 20

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 a_i ∈k.)

For eachm∈Z, we leth_m 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 hm = 0 whenm<0.)

Let J be the ideal ofP generated by thek differences hn−k+1−a1, hn−k+2−a2, . . . , hn−ak. 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.)

5 / 20

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 a_i ∈k.)

For eachm∈Z, we leth_m 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 hm = 0 whenm<0.)

Let J be the ideal ofP generated by thek differences hn−k+1−a1, hn−k+2−a2, . . . , hn−ak. 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.)

5 / 20

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.

6 / 20

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.

6 / 20

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.

6 / 20

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.

6 / 20

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.

6 / 20

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.

7 / 20

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.

7 / 20

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.

7 / 20

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).

8 / 20

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).

8 / 20

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).

8 / 20

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).

8 / 20

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).

8 / 20

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).

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Theh-basis

For eachν = (ν₁, ν₂, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set hν =hν1hν2· · ·hν` ∈ S.

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

λ∈P_k,n.

The transfer matrix between the two bases (s_λ)_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)^jak−js_(m,1^j₎, where m,1^j

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

| {z }

j ones

) (a hook-shaped partition).

9 / 20

Theh-basis

For eachν = (ν₁, ν₂, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set hν =hν1hν2· · ·hν` ∈ S.

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)^jak−js_(m,1^j₎, where m,1^j

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

| {z }

j ones

) (a hook-shaped partition).

9 / 20

Theh-basis

For eachν = (ν₁, ν₂, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set hν =hν1hν2· · ·hν` ∈ S.

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)^jak−js_(m,1^j₎,

where m,1^j

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

| {z }

jones

) (a hook-shaped partition).

9 / 20

Theh-basis

For eachν = (ν₁, ν₂, . . . , ν<sub>`</sub>)∈Z<sup>`</sup> (e.g., a partition), set hν =hν1hν2· · ·hν` ∈ S.

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)^jak−js_(m,1^j₎, where m,1^j

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

| {z }

jones

) (a hook-shaped partition).

9 / 20

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₍₎.

10 / 20

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₍₎.

10 / 20

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₍₎.

10 / 20

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₍₎.

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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.

11 / 20

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”?

12 / 20

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”?

12 / 20

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”?

12 / 20

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”?

12 / 20

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”? 12 / 20

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”? 12 / 20

S_k-module structure

The symmetric group S_k acts onP, with invariant ringS.

What is the S_k-module structure on PJ ?

Almost-theorem (G., needs to be checked): Assume that kis a Q-algebra. Then, as S_k-modules,

PJ ∼= PPS⁺

×

n k

∼=



 kS_k

|{z}

regular rep





×

n k

,

wherePS⁺ is the ideal of P generated by symmetric polynomials with constant term 0.

13 / 20

S_k-module structure

The symmetric group S_k acts onP, with invariant ringS.

What is the S_k-module structure on PJ ?

Almost-theorem (G., needs to be checked): Assume that kis a Q-algebra. Then, as S_k-modules,

PJ ∼= PPS⁺

×

n k

∼=



 kS_k

|{z}

regular rep





×

n k

,

wherePS⁺ is the ideal of P generated by symmetric polynomials with constant term 0.

13 / 20

Deforming symmetric functions, 1

Let us recall symmetricfunctions (not polynomials) now;

we’ll need them soon anyway.

S :={symmetric polynomials in x₁,x₂, . . . ,x_k}; Λ :={symmetric functions inx₁,x₂,x₃, . . .}. We use standard notations for symmetric functions, but in boldface:

e= elementary symmetric, h= complete homogeneous,

s= Schur (or skew Schur).

We have

S ∼= Λ(e_k+1, e_k+2, e_k+3, . . .)_ideal, thus SI ∼= Λ(hn−k+1−a₁, hn−k+2−a₂, . . . , h_n−a_k,

e_k+1, e_k+2, e_k+3, . . .)_ideal. So why not replace the ej byej −bj too?

14 / 20

Deforming symmetric functions, 1

Let us recall symmetricfunctions (not polynomials) now;

we’ll need them soon anyway.

S :={symmetric polynomials in x₁,x₂, . . . ,x_k}; Λ :={symmetric functions inx₁,x₂,x₃, . . .}. We use standard notations for symmetric functions, but in boldface:

e= elementary symmetric, h= complete homogeneous,

s= Schur (or skew Schur).

We have

S ∼= Λ(e_k+1, e_k+2, e_k+3, . . .)_ideal, thus SI ∼= Λ(hn−k+1−a₁, hn−k+2−a₂, . . . , h_n−a_k,

e_k+1, e_k+2, e_k+3, . . .)_ideal.

So why not replace the ej byej −bj too?

14 / 20