€¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V...

Explicit Pieri Inclusions and Minimal Graded FreeResolutions of Modules of Covariants of SeveralVectors and Covectors for a General Linear Group

John A. Miller

Baylor University

Algebra SeminarTexas Tech UniversityFebruary 19, 2020

e-mail: john_miller5@baylor.eduurl: www.johnamiller.xyz

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 1

The following is joint work with Markus Hunziker and Mark Sepanskiof Baylor University.

Syzygies

James Sylvester

syz·y·gy /’sizije/

from the Greek συζυγoς [syzygos] meaning “yoked together”

Syzygies

R = C [z1, . . . , zn] polynomial ring

M =⟨g1, . . . , gb0

⟩fin. gen. R-module

Rb0 free R-module with basis {ei : 1 ≤ i ≤ b0}

Consider the R-module map ε : Rb0 →M given by ei 7→ gi. Then

Rb0ϵ−→M −→ 0 is exact.

An element of ker ε is called a relation or first syzygy of M .

ker ε is called the module of first syzygies.

Syzygies

M =⟨g1, . . . , gb0

Syzygies

M =⟨g1, . . . , gb0

Syzygies

The module of first syzygies, ker ε, is also a finitely generatedR-module.

Therefore, ∃ R-module map δ1 : Rb1 −→ Rb0 such that

Rb1δ1−→ Rb0

ϵ−→M −→ 0 is exact.

The module of second syzygies, ker δ1, is again finitely generated.

Therefore, ∃ R-module map δ2 : Rb2 −→ Rb1 such that

Rb2δ2−→ Rb1

δ1−→ Rb0ϵ−→M −→ 0 is exact.

etc.John A. Miller (Baylor University) Pieri Inclusions and Syzygies 5

Syzygies

Will this go on forever?

Syzygy Theorem, Hilbert 1890

For R = C [z1, . . . , zn] and M as above, ∃ exact sequence of freemodules (free resolution)

0 −→ Rbm δm−→ · · ·Rb1δ1−→ Rb0

ϵ−→M −→ 0,

where m ≤ n.

Classical Invariant Theory

H a group, W finite dimensional H-module

{x1, . . . , xn} coordinate functions of W

Definition

The coordinate ring C [W ] = C [x1, . . . , xn] is the (graded) C-algebra of polynomials from W to C.

The grading on C [W ] is given by

C [W ] =⊕d∈Z≥0

C [W ]d

where C [W ]dare the homog. poly. of degree d.

Classical Invariant Theory

H a group, W finite dimensional H-module

{x1, . . . , xn} coordinate functions of W

Definition

The coordinate ring C [W ] = C [x1, . . . , xn] is the (graded) C-algebra of polynomials from W to C.

The grading on C [W ] is given by

C [W ] =⊕d∈Z≥0

C [W ]d

where C [W ]dare the homog. poly. of degree d.

Invariants

Definition

f ∈ C [W ] is called invariant (or H-invariant) if f (h · w) = f (w)for all h ∈ H,w ∈ W , i.e. f is constant on orbits.

Definition

The subalgebra

C [W ]H:= {f ∈ C[W ] | f is H-invariant}

of C [W ] is called the ring of invariants.

Covariants

Definition

If U is another finite-dimensional H-representation, the moduleof covariants of W of type U is defined as the space

CovH(W,U) := {φ : Wpoly−→ U | φ (h · w) = h · φ (w)

∀ h ∈ H,w ∈ W} .

Note that if U = C is the trivial representation,

CovH(W,U) = C [W ]H.

Fundamental Problem of CIT

Arthur Cayley David Hilbert Hermann Weyl

Fundamental Problem of CIT

Find generators and syzygies (relations) for the ring of invariantsC [W ]

Hand, more generally, for modules of covariants.

A partial solution to this problem for C [W ]Hwas given by Weyl in

1939.John A. Miller (Baylor University) Pieri Inclusions and Syzygies 10

Weyls First Fundamental Theorem

H = GL (V ) , W = V ∗ ⊕ · · · ⊕ V ∗︸︷︷︸p copies

⊕V ⊕ · · · ⊕ V︸︷︷︸q copies

= (V ∗)p ⊕ V q

For 1 ≤ i ≤ p and 1 ≤ j ≤ q, define

fij : (V∗)

p ⊕ V q → C

byfij (λ1, . . . , λp, v1, . . . , vq) = λi (vj) .

H = GL (V ) , W = V ∗ ⊕ · · · ⊕ V ∗︸︷︷︸p copies

⊕V ⊕ · · · ⊕ V︸︷︷︸q copies

= (V ∗)p ⊕ V q

fij : (V∗)

p ⊕ V q → C

Then, by construction, fij ∈ C [(V ∗)p ⊕ V q].

H = GL (V ) , W = V ∗ ⊕ · · · ⊕ V ∗︸︷︷︸p copies

⊕V ⊕ · · · ⊕ V︸︷︷︸q copies

= (V ∗)p ⊕ V q

fij : (V∗)

p ⊕ V q → Cby

fij (λ1, . . . , λp, v1, . . . , vq) = λi (vj) .

In fact, fij ∈ C[(V ∗)p ⊕ V q]GL(V ):

fij (g· (λ1, . . . , λp, v1, . . . , vq))

= fij (g · λ1, . . . , g · λp, g · v1, . . . , g · vq)= fij (λ1g

−1, . . . , λpg−1, g · v1, . . . , g · vq)

= λi (g−1g) · vj

= λivj = fij (λ1, . . . , λp, v1, . . . , vq)

H = GL (V ) , W = V ∗ ⊕ · · · ⊕ V ∗︸︷︷︸p copies

⊕V ⊕ · · · ⊕ V︸︷︷︸q copies

= (V ∗)p ⊕ V q

fij : (V∗)

p ⊕ V q → C

Theorem (FFT for GL (V ), Weyl 1939)

The basic invariants fij generate the invariant ring

C [(V ∗)p ⊕ V q]

GL(V )as a C-algebra.

Weyls Second Fundamental Theorem

R := C [zij | 1 ≤ i ≤ p, 1 ≤ j ≤ q]

C-alg hom ε : R→ C [(V ∗)p ⊕ V q]

GL(V )given by

zij 7→ fij.

Then, by the FFT,

Rϵ−→ C [(V ∗)

p ⊕ V q]GL(V ) −→ 0

is exact.

Theorem (SFT for GL (V ), Weyl 1939)

The first syzygies of the invariant ring C[(V ∗)p ⊕ V q]GL(V ) are

generated by the (k+1)× (k+1) minors of the matrix z = (zij),where k = dimV .

Remark: The minimal graded free resolution of C [(V ∗)p ⊕ V q]

GL(V )

as an R-module was first computed by Lascoux in 1978.

Weyls Second Fundamental Theorem

R := C [zij | 1 ≤ i ≤ p, 1 ≤ j ≤ q]

C-alg hom ε : R→ C [(V ∗)p ⊕ V q]

GL(V )given by

zij 7→ fij.

Then, by the FFT,

Rϵ−→ C [(V ∗)

p ⊕ V q]GL(V ) −→ 0

is exact.

Theorem (SFT for GL (V ), Weyl 1939)

The first syzygies of the invariant ring C[(V ∗)p ⊕ V q]GL(V ) are

generated by the (k+1)× (k+1) minors of the matrix z = (zij),where k = dimV .

Remark: The minimal graded free resolution of C [(V ∗)p ⊕ V q]

GL(V )

as an R-module was first computed by Lascoux in 1978.John A. Miller (Baylor University) Pieri Inclusions and Syzygies 12

Main Results

Hunziker–M–Sepanski

In the context of Hermann Weyl’s FT of invariant theory. for theclassical groups GL (V ), O (V ), and Sp (V ), we are able to:

Compute the syzygies of all modules of covariantsuniformly

Describe the differentials explicitly

Via Howe duality: modules of covariants ←→ unitarizablehighest weight modules

These then have BGG resolutions in some parabolic category OHighest weight modules σ ⇝ “compressed” λ

Parametrize the corresponding minimal free resolution via λ

Parameters “unzip” to give Verma modules

Main Results

Via Howe duality: modules of covariants ←→ unitarizablehighest weight modules

These then have BGG resolutions in some parabolic category OHighest weight modules σ ⇝ “compressed” λ

Parametrize the corresponding minimal free resolution via λ

Parameters “unzip” to give Verma modules

Main Results

Visualization:

0 −→ −→ ↗↘⊕ ↘↗ −→ · −→M −→ 0,

Today: Explain the set-up in the case H = GL (V ) and give anexample.

Notation for GL (V ) Reps

H = GL (V ) where dim(V ) = k

{polynomial irreducible reps of H}xy bij

{λ = (λ1, . . . , λk) | λi ∈ Z, λ1 ≥ · · · ≥ λk ≥ 0}

λ = (λ1, . . . , λk) ⇝ Young diagram with λ1 boxes in the first row,

λ2 boxes in the second row, etc.

We will just write the Young diagram λ for the correspondingSchur-Weyl module Sλ (V ).

{λ = (λ1, . . . , λk) | λi ∈ Z, λ1 ≥ · · · ≥ λk ≥ 0}

λ = (4, 3, 1, 1) ⇝ Sλ (V )←→ λ =

A basis for Sλ (V ) is the set of semi-standard tableaux of shape λ.

That is, all fillings of λ with the alphabet [k] = {1, . . . , k} where thefilling is weakly increasing across rows and strictly increasing downcolumns.

E.g.1 1 1 12 2 234

1 2 3 32 3 434

Note that the tableau with all ones in the first row, all twos in thesecond row, etc. is the highest weight vector for Sλ (V ).

λ = (4, 3, 1, 1) ⇝ Sλ (V )←→ λ =

E.g.1 1 1 12 2 234

1 2 3 32 3 434

λ = (4, 3, 1, 1) ⇝ Sλ (V )←→ λ =

E.g.1 1 1 12 2 234

1 2 3 32 3 434

λ = (4, 3, 1, 1) ⇝ Sλ (V )←→ λ =

E.g.1 1 1 12 2 234

1 2 3 32 3 434

For general irreps of H = GL (V ) we have

H = {irreps of H}xy bij

{σ = (σ1, . . . , σk) | σi ∈ Z, σ1 ≥ · · · ≥ σk}

So, a general GL (V ) irrep is F (k)σ = Fσ for some

σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)︸︷︷︸k

E.g. σ = (3, 2, 2, 0, 0,−1,−4). We can associate to σ two Youngdiagrams:

σ+ = (3, 2, 2) = and σ− = (4, 1) =

{σ = (σ1, . . . , σk) | σi ∈ Z, σ1 ≥ · · · ≥ σk}

σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)︸︷︷︸k

σ+ = (3, 2, 2) = and σ− = (4, 1) =

{σ = (σ1, . . . , σk) | σi ∈ Z, σ1 ≥ · · · ≥ σk}

σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)︸︷︷︸k

σ+ = (3, 2, 2) = and σ− = (4, 1) =

Back to Covariants

H = GL (V ) , dim (V ) = k, W = (V ∗)p ⊕ V q

Let Σ ={σ ∈ H | CovGL(V ) ((V

∗)p ⊕ V q, Fσ) 6= 0

Theorem (Kashiwara-Verge 1978)

σ ∈ Σ ⇐⇒ σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)

with 0 ≤ i ≤ q, 0 ≤ j ≤ p

Furthermore, for each σ ∈ Σ we can associate a partition λ (σ)to CovGL(V ) ((V

∗)p ⊕ V q, Fσ), namely

λ (σ) = (−k, . . . ,−k,−k −mj, . . . ,−k −m1︸︷︷︸p

;n1, . . . , ni, 0, . . . , 0︸︷︷︸q

Furthermore, the map σ 7→ λ(σ) is injective.

Back to Covariants

H = GL (V ) , dim (V ) = k, W = (V ∗)p ⊕ V q

Let Σ ={σ ∈ H | CovGL(V ) ((V

∗)p ⊕ V q, Fσ) 6= 0

Theorem (Kashiwara-Verge 1978)

σ ∈ Σ ⇐⇒ σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)

with 0 ≤ i ≤ q, 0 ≤ j ≤ p

Furthermore, for each σ ∈ Σ we can associate a partition λ (σ)to CovGL(V ) ((V

∗)p ⊕ V q, Fσ), namely

λ (σ) = (−k, . . . ,−k,−k −mj, . . . ,−k −m1︸︷︷︸p

;n1, . . . , ni, 0, . . . , 0︸︷︷︸q

Furthermore, the map σ 7→ λ(σ) is injective.

Reduction of σ

σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)↓

λ (σ) = (−k, . . . ,−k,−k −mj, . . . ,−k −m1︸︷︷︸p

;n1, . . . , ni, 0, . . . , 0︸︷︷︸q

↓(λ+ ρ)

′= ( , . . . ,︸︷︷︸

; , . . . ,︸︷︷︸q′

The terms in the resolution for M = CovGL(V ) ((V∗)

p ⊕ V q, Fσ) willbe parametrized by all Young diagrams contained in a p′ × q′

rectangle.

0 −→ −→ ↗↘⊕ ↘↗ −→ · −→M −→ 0

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−k,−k,−k,−k −m2,−k −m1︸︷︷︸5

;n1, 0, 0, 0, 0︸︷︷︸5

↓λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

How to get from λ+ ρ to (λ+ ρ)′?

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−k,−k,−k,−k −m2,−k −m1︸︷︷︸5

;n1, 0, 0, 0, 0︸︷︷︸5

↓λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−4,−4,−4,−5,−8; 3, 0, 0, 0, 0)

↓λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−4,−4,−4,−5,−8; 3, 0, 0, 0, 0)↓

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−4,−4,−4,−5,−8; 3, 0, 0, 0, 0)↓

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

1 Delete repeats on left & right(5, 4, 3, 1,−3; 7, 3, 2, 1, 0

)= (5, 4,−3; 7, 2, 0)

2 Delete any numbers on left < all numbers on right(5, 4, −3; 7, 2, 0

)= (5, 4; 7, 2, 0)

3 Delete any numbers on right > all numbers on left(5, 4; 7, 2, 0

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

)= (5, 4,−3; 7, 2, 0)

)= (5, 4; 7, 2, 0)

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

)= (5, 4,−3; 7, 2, 0)

)= (5, 4; 7, 2, 0)

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

)= (5, 4,−3; 7, 2, 0)

)= (5, 4; 7, 2, 0)

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

)= (5, 4,−3; 7, 2, 0)

)= (5, 4; 7, 2, 0)

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ −→ ↗↘

⊕ ↘↗ −→ · −→ CovH(W,Fσ) −→ 0

Starting with the box on the bottom left, label diagonally viasuccessive differences in (λ+ ρ)

′= (5, 4; 2, 0):

5− 4 = 1 ,

4− 2 = 2 ,

2− 0 = 2

This “compressed” version needs to be unzipped.

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ −→ ↗↘

⊕ ↘↗ −→ · −→ CovH(W,Fσ) −→ 0

′= (5, 4; 2, 0):

5− 4 = 1 ,

4− 2 = 2 ,

2− 0 = 2

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ −→ ↗↘

⊕ ↘↗ −→ · −→ CovH(W,Fσ) −→ 0

′= (5, 4; 2, 0):

5− 4 = 1 ,

4− 2 = 2 ,

2− 0 = 2

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ 1 −→ 1

↗↘

↘↗ −→ · −→ CovH(W,Fσ) −→ 0

′= (5, 4; 2, 0):

5− 4 = 1 ,

4− 2 = 2 ,

2− 0 = 2

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ 21 2 −→

↗↘

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

′= (5, 4; 2, 0):

5− 4 = 1 , 4− 2 = 2 ,

2− 0 = 2

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ 2 21 2 −→

↗↘

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

′= (5, 4; 2, 0):

5− 4 = 1 , 4− 2 = 2 , 2− 0 = 2

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ 2 21 2 −→

↗↘

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

′= (5, 4; 2, 0):

5− 4 = 1 , 4− 2 = 2 , 2− 0 = 2

Example

Unzip 2 21 :

σ = (3, 0,−1,−4)⇝ σ+ = (3) = σ− = (4, 1) =

Add boxes to the columns of σ− and σ+ as prescribed by thecolored diagram and its transpose, respectively.

2 21 = C [Mp×q]⊗

Example

Unzip 2 21 :

σ = (3, 0,−1,−4)⇝ σ+ = (3) = σ− = (4, 1) =

2 21 = C [Mp×q]⊗

Example

Unzip 2 21 :

σ = (3, 0,−1,−4)⇝ σ+ = (3) = σ− = (4, 1) =

2 21 = C [Mp×q]⊗

Example

Unzip 2 21 :

σ = (3, 0,−1,−4)⇝ σ+ = (3) = σ− = (4, 1) =

2 21 = C [M5×5]⊗

Example

Then the maps in the resolution look like

2 21 → 2

which after unzipping looks like

C [M5×5]⊗

→ C [M5×5]⊗

Here, each tableau on the left is losing 2 boxes.

As a C [M5×5]-module map, it is enough to consider ⊗

→ C [M5×5]2 ⊗

Example

2 21 → 2

C [M5×5]⊗

→ C [M5×5]2 ⊗

Example

2 21 → 2

C [M5×5]⊗

→ C [M5×5]2 ⊗

Example

→ C [M5×5]2 ⊗

As a GL(5;C)×GL(5;C) rep, C [M5×5]2 decomposes into

C [M5×5]2 =(

⊗)⊕(⊗

)In general, C [M5×5]d decomposes as above with all shapes thathave d boxes and ≤ 5 rows.

Then, we end up with...

Example

→ C [M5×5]2 ⊗

C [M5×5]2 =(

⊗)⊕(⊗

Example

→ (⊗

C [M5×5]2 =(

⊗)⊕(⊗

Example

→ (⊗

Computing the syzygies for CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

comes down to finding C[M5×5]-module and GL(V )-equivariantPieri inclusions, such as

→ ⊗ and → ⊗

For the rest of this talk: updated description of Pieri inclusions.

Example

→ (⊗

Computing the syzygies for CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

comes down to finding C[M5×5]-module and GL(V )-equivariantPieri inclusions, such as

→ ⊗ and → ⊗

For the rest of this talk: updated description of Pieri inclusions.

The Pieri Rule

Theorem (Pieri Rule)

Let µ be a partition corresponding to a Schur-Weyl module Sµ (V )and ν = (1, . . . , 1) be a partition of m. Then we have an isomor-phism of GL (V )-modules

Sν (V )⊗ Sµ (V ) ∼=⊕λ

Sλ (V )

where the sum is over all λ ⊃ µ obtained by adding m boxes toµ with no two boxes in the same row. Similarly,

S(m) (V )⊗ Sµ (V ) ∼=⊕λ

Sλ (V )

where the sum is over all λ ⊃ µ obtained by adding m boxes toµ with no two boxes in the same column.

The Pieri Rule - One Box

⊗ ∼= ⊕ ⊕

⊗ ∼= ⊕ ⊕ ⊕

The Pieri Rule - Two Boxes

⊗ ∼= ⊕ ⊕ ⊕ ⊕

⊗ ∼= ⊕ ⊕ ⊕

The Pieri Rule - Three Boxes

⊗ ∼= ⊕ ⊕ ⊕ ⊕ ⊕

⊗ ∼= ⊕ ⊕ ⊕

Pieri Inclusions

From the Pieri rule we get maps

Sλ (V )→ Sν (V )⊗ Sµ (V ) ,

called Pieri inclusions, unique up to non-zero scalar multiple.

−→ ⊗ −→ ⊗

These Pieri inclusions were first given by Olver in his thesis (1982)via iterating one box removal, and made more explicit by Sam(2009) and Sam and Weyman (2012).

We give a general closed form description of Pieri inclusionsremoving m boxes and show this rule is more efficient.

Pieri Inclusions

From the Pieri rule we get maps

Sλ (V )→ Sν (V )⊗ Sµ (V ) ,

called Pieri inclusions, unique up to non-zero scalar multiple.

−→ ⊗ −→ ⊗

These Pieri inclusions were first given by Olver in his thesis (1982)via iterating one box removal, and made more explicit by Sam(2009) and Sam and Weyman (2012).

We give a general closed form description of Pieri inclusionsremoving m boxes and show this rule is more efficient.

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ ,

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ 1

21 ⊗ 3

ΦO =∑J

(−1)|J|JcJ

E.g.,ΦO−→ ⊗ , ΦO

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ 1

21 ⊗ 3

ΦO =∑J

(−1)|J|JcJ

E.g.,ΦO−→ ⊗ , ΦO

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ 1

21 ⊗ 3

ΦO =∑J

(−1)|J|JcJ

E.g.,ΦO−→ ⊗ , ΦO

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ 1

21 ⊗ 3

ΦO =∑J

(−1)|J|JcJ

E.g.,ΦO−→ ⊗ , ΦO

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ 1

21 ⊗ 3

ΦO =∑J

(−1)|J|JcJ

E.g.,ΦO−→ ⊗ , ΦO

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ 1

21 ⊗ 3

ΦO =∑J

(−1)|J|JcJ

E.g.,ΦO−→ ⊗ , ΦO

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ 1

21 ⊗ 3

ΦO =∑J

(−1)|J|JcJ

E.g.,ΦO−→ ⊗ , ΦO

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ −1

21 ⊗ 2

ΦO =∑J

(−1)|J|JcJ

E.g.,ΦO−→ ⊗ , ΦO

)= − 3 ⊗ 1

2 + 2 ⊗ 13 − 1 ⊗ 2

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ −1

21 ⊗ 2

ΦO =∑J

(−1)|J|JcJ

E.g.,ΦO−→ ⊗ , ΦO

)= − 3 ⊗ 1

2 + 2 ⊗ 13 − 1 ⊗ 2

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ −1

21 ⊗ 2

1 ⇝ −1

21 ⊗ 2

Blocks of a diagram

Block notation: λ = (wh11 , w

h22 . . . , w

hN−1N−1 , w

hNN ) where wi < wi+1

and each wi appears as a part of λ exactly hi times.

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

Blocks of a diagram

h22 . . . , w

hN−1N−1 , w

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

Blocks of a diagram

h22 . . . , w

hN−1N−1 , w

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

Blocks of a diagram

h22 . . . , w

hN−1N−1 , w

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

Blocks of a diagram

h22 . . . , w

hN−1N−1 , w

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

Blocks of a diagram

h22 . . . , w

hN−1N−1 , w

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

Blocks of a diagram

h22 . . . , w

hN−1N−1 , w

wN−1

hN−1

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

Blocks of a diagram

h22 . . . , w

hN−1N−1 , w

hN (1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

New description of Pieri inclusions

Φ =∑P

(−1)|P |P

The P are the ways to remove the indicated box up and out of thediagram with a row skipping restriction. The coefficients H(P )are similar to the cJ , but depend only on the blocks used.

E.g.,Φ−→ ⊗ ,

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ − 1 ⊗ 23

Φ =∑P

(−1)|P |P

E.g.,Φ−→ ⊗ , Φ

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ − 1 ⊗ 23

Φ =∑P

(−1)|P |P

E.g.,Φ−→ ⊗ , Φ

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ − 1 ⊗ 23

Φ =∑P

(−1)|P |P

E.g.,Φ−→ ⊗ , Φ

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ − 1 ⊗ 23

Φ =∑P

(−1)|P |P

E.g.,Φ−→ ⊗ , Φ

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ − 1 ⊗ 23

Φ =∑P

(−1)|P |P

E.g.,Φ−→ ⊗ , Φ

)= − 3 ⊗ 1

2 + 2 ⊗ 13 − 1 ⊗ 2

1 ⇝ − 3 ⊗ 12 ,

1 ⇝ 2 ⊗ 13

1 ⇝ − 1 ⊗ 23

Pieri inclusions removing two boxes

↪→ ⊗ : 7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

65432 41 2

⇝ 56 ⊗

1 22 434

65432 41 2

⇝ 34 ⊗

1 22 456

65432 41 2

⇝ −1

434 ⊗

1 22 456

65432 41 2

⇝ −1

431 ⊗

4 12 645

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

65432 41 2

⇝ 56 ⊗

1 22 434

65432 41 2

⇝ 34 ⊗

1 22 456

65432 41 2

⇝ −1

434 ⊗

1 22 456

65432 41 2

⇝ −1

431 ⊗

4 12 645

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

65432 41 2

⇝ 56 ⊗

1 22 434

65432 41 2

⇝ 34 ⊗

1 22 456

65432 41 2

⇝ −1

434 ⊗

1 22 456

65432 41 2

⇝ −1

431 ⊗

4 12 645

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

65432 41 2

⇝ 56 ⊗

1 22 434

65432 41 2

⇝ 34 ⊗

1 22 456

65432 41 2

⇝ −1

434 ⊗

1 22 456

65432 41 2

⇝ −1

431 ⊗

4 12 645

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

65432 41 2

⇝ 56 ⊗

1 22 434

65432 41 2

⇝ 34 ⊗

1 22 456

65432 41 2

⇝ −1

434 ⊗

1 22 456

65432 41 2

⇝ −1

431 ⊗

4 12 645

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

65432 41 2

⇝ 56 ⊗

1 22 434

65432 41 2

⇝ 34 ⊗

1 22 456

65432 41 2

⇝ −1

434 ⊗

1 22 456

65432 41 2

⇝ −1

431 ⊗

4 12 645

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

65432 41 2

⇝ 56 ⊗

1 22 434

65432 41 2

⇝ 34 ⊗

1 22 456

65432 41 2

⇝ −1

434 ⊗

1 22 456

65432 41 2

413 ⊗

1 42 645

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

65432 41 2

⇝ 56 ⊗

1 22 434

65432 41 2

⇝ 34 ⊗

1 22 456

65432 41 2

⇝ −1

434 ⊗

1 22 456

65432 41 2

413 ⊗

1 42 645

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

65432 41 2

⇝ 56 ⊗

1 22 434

65432 41 2

⇝ 34 ⊗

1 22 456

65432 41 2

⇝ −1

434 ⊗

1 22 456

65432 41 2

413 ⊗

1 42 645

65432 41 2

2023 ⊗

1 34 456

1023 ⊗

1 43 546

Complexity of the descriptions

In both the old and new descriptions, the number of terms in thePieri inclusion acting on λ = (w

h11 , . . . , w

hNN ) depends on the

number of paths acting on the diagram.

Old description: paths given by the number of choices of rows inthe diagram, where you must choose the first (bottom most) row.

2h1−1 ·N∏i=2

New description: paths given by the number of choices of rows inthe diagram, where you must choose the first row and cannot skiprows within blocks.

h1 ·N∏i=2

(hi + 1) .

h11 , . . . , w

2h1−1 ·N∏i=2

h1 ·N∏i=2

(hi + 1) .

h11 , . . . , w

2h1−1 ·N∏i=2

h1 ·N∏i=2

(hi + 1) .

Computation Time in Macaulay2

Old description:

New description:

Old description:

New description:

Old description:

New description:

Back to the syzygies example

Recall:

W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

0 −→ 2 21 2 −→

↗↘

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

And after unzipping, the maps look like ⊗

→ (⊗

and are induced by Pieri inclusions.

Consider the image of the highest weight vector:

5 5 5 54 43 321

⊗1 1 12 2345

We will apply the Pieri inclusion map removing two boxes to eachterm separately, then put them together.

1 1 12 2345

2-paths P

(−1)|P |

H (P )Pout

( )⊗ P

Some terms in the sum are (pre-straightening):

⇝ 45 ⊗

1 1 12 23

(13 ⊗

1 4 12 25

⇝ − 1

(31 ⊗

2 1 12 54

1 1 12 2345

2-paths P

(−1)|P |

H (P )Pout

( )⊗ P

⇝ 45 ⊗

1 1 12 23

(13 ⊗

1 4 12 25

⇝ − 1

(31 ⊗

2 1 12 54

1 1 12 2345

= 45 ⊗

1 1 12 23

− 35 ⊗

1 1 12 24

(25 ⊗

1 1 12 43

(25 ⊗

1 1 12 34

)− 3

(15 ⊗

1 1 42 23

)− 2

(15 ⊗

1 1 22 34

(15 ⊗

1 1 22 43

(15 ⊗

1 1 32 24

4 ⊗1 1 12 25

+ · · ·

5 5 5 54 43 321

2-paths P

(−1)|P |

H (P )Pout

( )⊗ P

345 5 5

⇝ 43 ⊗

5 5 5 54321

345 5 5

⇝ −1

53 ⊗

4 5 5 54321

345 5 5

⇝ −1

45 ⊗

5 5 3 54321

5 5 5 54 43 321

2-paths P

(−1)|P |

H (P )Pout

( )⊗ P

345 5 5

⇝ 43 ⊗

5 5 5 54321

345 5 5

⇝ −1

53 ⊗

4 5 5 54321

345 5 5

⇝ −1

45 ⊗

5 5 3 54321

5 5 5 54 43 321

43 ⊗

5 5 5 54321

− 16

53 ⊗

5 5 5 44321

54 ⊗

5 5 5 34321

5 5 5 54 43 321

⊗1 1 12 2345y

∑2-paths P,2-paths Q

(−1)|P |+|Q|

H(P)H (Q)

E.g. one term in the image is

(43 ⊗

5 5 5 54321

⊗1 1 22 34

∈ C[M5×5]2 ⊗

Still need 43 ⊗

15 ∈ C[M5×5]2, given by a minor:

z11 z12 z13 z14 z15z21 z22 z23 z24 z25z31 z32 z33 z34 z35z41 z42 z43 z44 z45z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

Still need 43 ⊗

z11 z12 z13 z14 z15z21 z22 z23 z24 z25

3 z31 z32 z33 z34 z354 z41 z42 z43 z44 z45

z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

Still need 43 ⊗

z11 z12 z13 z14 z15z21 z22 z23 z24 z25

3 z31 z32 z33 z34 z354 z41 z42 z43 z44 z45

z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

Still need 43 ⊗

z11 z12 z13 z14 z15z21 z22 z23 z24 z25

3 z31 z32 z33 z34 z354 z41 z42 z43 z44 z45

z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

Still need 43 ⊗

z11 z12 z13 z14 z15z21 z22 z23 z24 z25

3 z31 z32 z33 z34 z354 z41 z42 z43 z44 z45

z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

“Unzip” each term.

Compute the differentials similarly.

Generally, for any W = (V ∗)p ⊕ V q and any σ.

0 −→ 2 21 2 −→

↗↘

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

References

Peter Olver, Differential Hyperforms I, University of MinnesotaMathematics Report, available www.math.umn.edu/~olver/Steven Sam, Computing inclusions of Schur modules, Journal ofSoftware for Algebra and Geometry, arXiv:0810.4666

Steven Sam, Jerzy Weyman, Pieri resolutions for classical groups,Journal of Algebra, arXiv:0907.4505

Markus Hunziker, John Miller, Mark Sepanski, Explicit PieriInclusions, arXiv:1911.11045

Markus Hunziker, John Miller, Mark Sepanski, Minimal GradedFree Resolutions of Modules of Covariants For Classical Groups, comingsoon

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