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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: [email protected]: 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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 2
Syzygies
James Sylvester
syz·y·gy /’sizije/
from the Greek συζυγoς [syzygos] meaning “yoked together”
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 3
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 4
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 4
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 4
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 6
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 7
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 7
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 8
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 9
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) .
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 11
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) .
Then, by construction, fij ∈ C [(V ∗)p ⊕ V q].
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 11
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 → 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)
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 11
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) .
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 11
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
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
Idea:
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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 13
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
Idea:
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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 13
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
Visualization:
0 −→ −→ ↗↘⊕ ↘↗ −→ · −→M −→ 0,
Today: Explain the set-up in the case H = GL (V ) and give anexample.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 13
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 ).
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 14
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 ).
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 14
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 ).
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 14
Notation for GL (V ) Reps
E.g.
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 ).
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 15
Notation for GL (V ) Reps
E.g.
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 ).
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 15
Notation for GL (V ) Reps
E.g.
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 ).
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 15
Notation for GL (V ) Reps
E.g.
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 ).
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 15
Notation for GL (V ) Reps
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) =
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 16
Notation for GL (V ) Reps
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) =
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 16
Notation for GL (V ) Reps
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) =
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 16
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 17
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 17
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)
′= ( , . . . ,︸ ︷︷ ︸
p′
; , . . . ,︸ ︷︷ ︸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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 18
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)′?
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19
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)′?
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19
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).
How to get from λ+ ρ to (λ+ ρ)′?
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19
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).
How to get from λ+ ρ to (λ+ ρ)′?
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19
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).
How to get from λ+ ρ to (λ+ ρ)′?
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21
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
↗↘
⊕1
↘↗ −→ · −→ 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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21
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 −→
21
↗↘
2
⊕21
↘↗ 2 −→ · −→ 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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21
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 21
↗↘
2 2
⊕21
↘↗ 2 −→ · −→ 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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21
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 21
↗↘
2 2
⊕21
↘↗ 2 −→ · −→ 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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21
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]⊗
⊗
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 22
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]⊗
⊗
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 22
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]⊗
⊗
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 22
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 [M5×5]⊗
⊗
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 22
Example
Then the maps in the resolution look like
2 21 → 2
1 ,
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 ⊗
⊗
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 23
Example
Then the maps in the resolution look like
2 21 → 2
1 ,
which after unzipping looks like
C [M5×5]⊗
⊗
→ C [M5×5]2 ⊗
⊗
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 ⊗
⊗
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 23
Example
Then the maps in the resolution look like
2 21 → 2
1 ,
which after unzipping looks like
C [M5×5]⊗
⊗
→ C [M5×5]2 ⊗
⊗
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 ⊗
⊗
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 23
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...
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24
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...
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24
Example
⊗
→ (⊗
)⊗
⊗
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...
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24
Example
⊗
→ (⊗
)⊗
⊗
Computing the syzygies for CovGL(V )
((V ∗)
5 ⊕ V 5, F (4)σ
)then
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24
Example
⊗
→ (⊗
)⊗
⊗
Computing the syzygies for CovGL(V )
((V ∗)
5 ⊕ V 5, F (4)σ
)then
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 25
The Pieri Rule - One Box
⊗ ∼= ⊕ ⊕
⊗ ∼= ⊕ ⊕ ⊕
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 26
The Pieri Rule - Two Boxes
⊗ ∼= ⊕ ⊕ ⊕ ⊕
⊗ ∼= ⊕ ⊕ ⊕
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 27
The Pieri Rule - Three Boxes
⊗ ∼= ⊕ ⊕ ⊕ ⊕ ⊕
⊗ ∼= ⊕ ⊕ ⊕
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 28
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.
E.g.
−→ ⊗ −→ ⊗
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 29
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.
E.g.
−→ ⊗ −→ ⊗
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 29
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−→ ⊗ ,
ΦO
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,3
2
1 ⇝ 1
21 ⊗ 3
2
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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−→ ⊗ , ΦO
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,3
2
1 ⇝ 1
21 ⊗ 3
2
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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−→ ⊗ , ΦO
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,3
2
1 ⇝ 1
21 ⊗ 3
2
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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−→ ⊗ , ΦO
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,3
2
1 ⇝ 1
21 ⊗ 3
2
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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−→ ⊗ , ΦO
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,
3
2
1 ⇝ 1
21 ⊗ 3
2
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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−→ ⊗ , ΦO
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,3
2
1 ⇝ 1
21 ⊗ 3
2
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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−→ ⊗ , ΦO
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,3
2
1 ⇝ 1
21 ⊗ 3
2
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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−→ ⊗ , ΦO
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,3
2
1 ⇝ −1
21 ⊗ 2
3
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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−→ ⊗ , ΦO
(123
)= − 3 ⊗ 1
2 + 2 ⊗ 13 − 1 ⊗ 2
3
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,3
2
1 ⇝ −1
21 ⊗ 2
3
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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−→ ⊗ , ΦO
(123
)= − 3 ⊗ 1
2 + 2 ⊗ 13 − 1 ⊗ 2
3
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ −1
21 ⊗ 2
3 ,3
2
1 ⇝ −1
21 ⊗ 2
3
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30
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):
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31
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):
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31
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):
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31
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):
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31
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.
w1h1
...
(1, 1) = (12):
(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31
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.
w2h2
...
(1, 1) = (12):
(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31
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.
...
wN−1
hN−1
(1, 1) = (12):
(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31
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.
...
wN
hN (1, 1) = (12):
(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31
New description of Pieri inclusions
Φ =∑P
(−1)|P |P
H(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−→ ⊗ ,
Φ
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ − 1 ⊗ 23
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32
New description of Pieri inclusions
Φ =∑P
(−1)|P |P
H(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−→ ⊗ , Φ
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ − 1 ⊗ 23
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32
New description of Pieri inclusions
Φ =∑P
(−1)|P |P
H(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−→ ⊗ , Φ
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ − 1 ⊗ 23
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32
New description of Pieri inclusions
Φ =∑P
(−1)|P |P
H(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−→ ⊗ , Φ
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ − 1 ⊗ 23
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32
New description of Pieri inclusions
Φ =∑P
(−1)|P |P
H(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−→ ⊗ , Φ
(123
)=?
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ − 1 ⊗ 23
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32
New description of Pieri inclusions
Φ =∑P
(−1)|P |P
H(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−→ ⊗ , Φ
(123
)= − 3 ⊗ 1
2 + 2 ⊗ 13 − 1 ⊗ 2
3
3
2
1 ⇝ − 3 ⊗ 12 ,
3
2
1 ⇝ 2 ⊗ 13
3
2
1 ⇝ − 1 ⊗ 23
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32
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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33
Pieri inclusions removing two boxes
↪→ ⊗ :
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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33
Pieri inclusions removing two boxes
↪→ ⊗ :
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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33
Pieri inclusions removing two boxes
↪→ ⊗ :
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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33
Pieri inclusions removing two boxes
↪→ ⊗ :
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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33
Pieri inclusions removing two boxes
↪→ ⊗ :
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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33
Pieri inclusions removing two boxes
↪→ ⊗ :
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
413 ⊗
1 42 645
,
65432 41 2
⇝ − 1
5 · 423 ⊗
1 54 436
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33
Pieri inclusions removing two boxes
↪→ ⊗ :
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
413 ⊗
1 42 645
,
65432 41 2
⇝ − 1
5 · 423 ⊗
1 54 436
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33
Pieri inclusions removing two boxes
↪→ ⊗ :
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
413 ⊗
1 42 645
,
65432 41 2
⇝ 1
2023 ⊗
1 34 456
− 1
1023 ⊗
1 43 546
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33
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
2hi ,
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) .
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 34
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
2hi ,
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) .
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 34
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
2hi ,
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) .
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 34
Computation Time in Macaulay2
Old description:
New description:
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 35
Computation Time in Macaulay2
Old description:
New description:
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 35
Computation Time in Macaulay2
Old description:
New description:
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 35
Back to the syzygies example
Recall:
W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)
0 −→ 2 21 2 −→
2 21
↗↘
2 2
⊕21
↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0
And after unzipping, the maps look like ⊗
→ (⊗
)⊗
⊗
and are induced by Pieri inclusions.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 36
Back to the syzygies example
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 37
Back to the syzygies example
Φ
1 1 12 2345
=∑
2-paths P
(−1)|P |
H (P )Pout
( )⊗ P
( )
Some terms in the sum are (pre-straightening):
54321
21 1
⇝ 45 ⊗
1 1 12 23
,
54321
21 1
⇝ 1
5
(13 ⊗
1 4 12 25
)
54321
21 1
⇝ − 1
15
(31 ⊗
2 1 12 54
)
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 38
Back to the syzygies example
Φ
1 1 12 2345
=∑
2-paths P
(−1)|P |
H (P )Pout
( )⊗ P
( )
Some terms in the sum are (pre-straightening):
54321
21 1
⇝ 45 ⊗
1 1 12 23
,
54321
21 1
⇝ 1
5
(13 ⊗
1 4 12 25
)
54321
21 1
⇝ − 1
15
(31 ⊗
2 1 12 54
)
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 38
Back to the syzygies example
Φ
1 1 12 2345
= 45 ⊗
1 1 12 23
− 35 ⊗
1 1 12 24
− 2
3
(25 ⊗
1 1 12 43
)2
3
(25 ⊗
1 1 12 34
)− 3
5
(15 ⊗
1 1 42 23
)− 2
5
(15 ⊗
1 1 22 34
)+
2
5
(15 ⊗
1 1 22 43
)+
3
5
(15 ⊗
1 1 32 24
)+ 3
4 ⊗1 1 12 25
+ · · ·
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 39
Back to the syzygies example
Φ
5 5 5 54 43 321
=∑
2-paths P
(−1)|P |
H (P )Pout
( )⊗ P
Some terms in the sum are (pre-straightening):
12345
345 5 5
⇝ 43 ⊗
5 5 5 54321
,
12345
345 5 5
⇝ −1
3
53 ⊗
4 5 5 54321
12345
345 5 5
⇝ −1
3
45 ⊗
5 5 3 54321
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 40
Back to the syzygies example
Φ
5 5 5 54 43 321
=∑
2-paths P
(−1)|P |
H (P )Pout
( )⊗ P
Some terms in the sum are (pre-straightening):
12345
345 5 5
⇝ 43 ⊗
5 5 5 54321
,
12345
345 5 5
⇝ −1
3
53 ⊗
4 5 5 54321
12345
345 5 5
⇝ −1
3
45 ⊗
5 5 3 54321
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 40
Back to the syzygies example
Φ
5 5 5 54 43 321
= 4
43 ⊗
5 5 5 54321
− 16
3
53 ⊗
5 5 5 44321
+
16
3
54 ⊗
5 5 5 34321
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 41
Back to the syzygies example
5 5 5 54 43 321
⊗1 1 12 2345y
∑2-paths P,2-paths Q
(−1)|P |+|Q|
H(P)H (Q)
(⊗
)⊗
P
⊗Q
E.g. one term in the image is
8
5
(43 ⊗
15
)⊗
5 5 5 54321
⊗1 1 22 34
∈ C[M5×5]2 ⊗
⊗
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 42
Back to the syzygies example
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43
Back to the syzygies example
Still need 43 ⊗
15 ∈ C[M5×5]2, given by a minor:
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43
Back to the syzygies example
Still need 43 ⊗
15 ∈ C[M5×5]2, given by a minor:
1 5
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43
Back to the syzygies example
Still need 43 ⊗
15 ∈ C[M5×5]2, given by a minor:
1 5
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43
Back to the syzygies example
Still need 43 ⊗
15 ∈ C[M5×5]2, given by a minor:
1 5
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.
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43
“Unzip” each term.
Compute the differentials similarly.
Generally, for any W = (V ∗)p ⊕ V q and any σ.
0 −→ 2 21 2 −→
2 21
↗↘
2 2
⊕21
↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 44
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
John A. Miller (Baylor University) Pieri Inclusions and Syzygies 45