Invariant Theory for Quiverssites.lsa.umich.edu/.../2018/09/AuslanderSlides.pdf · Invariant Theory for Quivers Harm Derksen University of Michigan ... G homogeneous of degree d,

Invariant Theory for Quivers

Harm Derksen

University of Michigan

Maurice Auslander Distinguished Lecturesand International Conference

April 29, 2017

Harm Derksen Invariant Theory for Quivers

Invariant Theory

K = K algebraically closed fieldG reductive algebraic group (e.g., GLn, semi-simple, finite,. . . )V n-dimensional representation of GK [V ] ring of polynomial functions on V

G acts on K [V ]: for g ∈ G , f ∈ K [V ], g · f defined by

(g · f )(v) = f (g−1v), v ∈ V

Definition

K [V ]G = {f ∈ K [V ] | ∀g ∈ G g · f = f } invariant ring

Theorem (Hilbert 1890, Nagata 1963/Haboush 1975)

K [V ]G is a finitely generated K -algebra


Invariant Theory



(g · f )(v) = f (g−1v), v ∈ V

Definition





Invariant Theory



(g · f )(v) = f (g−1v), v ∈ V

Definition





Invariant Theory



(g · f )(v) = f (g−1v), v ∈ V

Definition





Example

Example

G = Sn acts on V = Kn by permutationsG acts on K [V ] = K [x1, . . . , xn] by permuting variables

K [V ]Sn = K [e1, e2, . . . , en] where

ek =∑

i1<i2<···<ik

xi1xi2 · · · xik

is k-th elementary symmetric function


Example

Example

G = Sn acts on V = Kn by permutationsG acts on K [V ] = K [x1, . . . , xn] by permuting variablesK [V ]Sn = K [e1, e2, . . . , en] where

ek =∑

i1<i2<···<ik

xi1xi2 · · · xik

is k-th elementary symmetric function


Geometric Invariant Theory

inclusion K [V ]G ↪→ K [V ] corresponds to a quotient

π : V → V //G

where V //G = SpecK [V ]G

Theorem

(a) π is surjective(b) for y ∈ V //G , π−1(y) contains exactly 1 closed orbit, say G · z(c) z lies in the close of each orbit in π−1(y)

Definition (Hilbert’s Nullcone)

N = π−1π(0) is Hilbert’s nullconeN = {v ∈ V | f (v) = 0 for all nonconst. homogen. f ∈ K [V ]G}N = {v ∈ V | 0 ∈ G · v}




π : V → V //G


Theorem







π : V → V //G


Theorem





Examples

Example

G = GL1 = K ? acts on V = K 3:

t · (x , y , z) = (tx , t3y , t−2z)

K [V ] = K [x , y , z ]

K [V ]G = K [x2z , xyz2, y2z3] ∼= K [a, b, c]/(b2 − ac)

π : K 3 → C := {(a, b, c) | b2 − ac}

N = {x = y = 0} ∪ {z = 0}


Examples

Example


t · (x , y , z) = (tx , t3y , t−2z)

K [V ] = K [x , y , z ]


π : K 3 → C := {(a, b, c) | b2 − ac}

N = {x = y = 0} ∪ {z = 0}


Examples

Example


t · (x , y , z) = (tx , t3y , t−2z)

K [V ] = K [x , y , z ]


π : K 3 → C := {(a, b, c) | b2 − ac}

N = {x = y = 0} ∪ {z = 0}


Examples

Example


t · (x , y , z) = (tx , t3y , t−2z)

K [V ] = K [x , y , z ]


π : K 3 → C := {(a, b, c) | b2 − ac}

N = {x = y = 0} ∪ {z = 0}



Construction of quotient of P(V ) with respect to G :

Definition

(a) v ∈ V is semistable if and only if 0 6∈ G · v(b) v ∈ V is stable if and only if v is semistable and

dimG · v = dimG

V ss semistable pointsV s stable points, open subset of V ss

P(V s) ⊆ P(V ss) ⊆ P(V )

Theorem

(a) π : P(V ss)→ P(V //G ) is a “good” quotient

(b) restriction to P(V s) is geometric quotient (orbits=fibers)




Definition


dimG · v = dimG


P(V s) ⊆ P(V ss) ⊆ P(V )

Theorem






Definition


dimG · v = dimG


P(V s) ⊆ P(V ss) ⊆ P(V )

Theorem




Constructive Invariant Theory

char(K ) = 0r = dimK [V ]G ≤ dimV = n

Theorem (Popov 1981)

if N ⊆ V is the zero set of f1, f2, . . . , fs ∈ K [V ]G homogeneous ofdegree d , then K [V ]G generated by invariants of degree ≤ rd .

Theorem (D. 2001)

if N ⊆ V is the zero set of f1, f2, . . . , fs ∈ K [V ]G homogeneous ofdegree ≤ d , then K [V ]G generated by invariants of degree≤ max{d , 38 rd

2}.

(improves bound r lcm(1, 2, . . . , d) of Popov.)


Constructive Invariant Theory

char(K ) = 0r = dimK [V ]G ≤ dimV = n

Theorem (Popov 1981)

if N ⊆ V is the zero set of f1, f2, . . . , fs ∈ K [V ]G homogeneous ofdegree d , then K [V ]G generated by invariants of degree ≤ rd .

Theorem (D. 2001)

if N ⊆ V is the zero set of f1, f2, . . . , fs ∈ K [V ]G homogeneous ofdegree ≤ d , then K [V ]G generated by invariants of degree≤ max{d , 38 rd

2}.

(improves bound r lcm(1, 2, . . . , d) of Popov.)


Quiver Representations

Definition

A quiver is a 4-tuple Q = (Q0,Q1, h, t), whereQ0, finite set of verticesQ1, finite set of arrowsh, t : Q1 → Q0

h(a) = ha head of arrow at(a) = ta tail of arrow a

Fix a field K

Definition

A quiver representation V (over K ) isfinite dimensional K -vector spaces V (x), x ∈ Q0, together withK -linear maps V (a) : V (ta)→ V (ha), a ∈ Q1


Quiver Representations

Definition

A quiver is a 4-tuple Q = (Q0,Q1, h, t), whereQ0, finite set of verticesQ1, finite set of arrowsh, t : Q1 → Q0

h(a) = ha head of arrow at(a) = ta tail of arrow a

Fix a field K

Definition

A quiver representation V (over K ) isfinite dimensional K -vector spaces V (x), x ∈ Q0, together withK -linear maps V (a) : V (ta)→ V (ha), a ∈ Q1


Representation Spaces

The dimension vector of a representation V is the function

dimV : x ∈ Q0 7→ dimV (x)

NQ0 is the set of dimension vectors

if V has dimension vector α ∈ NQ0 and we choose basis ofV (x) ∼= Kα(x), x ∈ Q0, then V (a) is α(ha)× α(ta) matrix fora ∈ Q1

We can view V as an element in the representation space

V = (V (a), a ∈ Q1) ∈ Repα(Q) :=∏a∈Q1

Hom(Kα(ta),Kα(ha))








V = (V (a), a ∈ Q1) ∈ Repα(Q) :=∏a∈Q1









V = (V (a), a ∈ Q1) ∈ Repα(Q) :=∏a∈Q1



Rings of Invariants for Loop Quivers

Definition

I(Q, α) = K [Repα(Q)]GLα invariant ring for quiver representations

Special case Q quiver with 1 vertex, m-loops, α = (p)Repα(Q) = Matp(K )m, GLα = GLp acts by conjugation

Theorem (Procesi)

If char(K ) = 0, then K [Matmn,n]GLn is generated by all

(A1,A2, . . . ,Am) 7→ Tr(Ai1Ai2 · · ·Aid )

Theorem (Razmyslov)

If char(K ) = 0, then K [Matmn,n]GLn is generated by invariants ofdegree ≤ n2



Definition



Theorem (Procesi)


(A1,A2, . . . ,Am) 7→ Tr(Ai1Ai2 · · ·Aid )

Theorem (Razmyslov)




Definition



Theorem (Procesi)


(A1,A2, . . . ,Am) 7→ Tr(Ai1Ai2 · · ·Aid )

Theorem (Razmyslov)



Rings of Invariants for Arbitrary Quivers

Q arbitary quiverif V representation, and p = akak−1 · · · a1 path, thenV (p) := V (ak)V (ak−1) · · ·V (a1)If p1, p2, . . . , pr paths (same head/tail), λ1, . . . , λr ∈ K , thenV (∑r

i=1 λipi ) :=∑r

i=1 λiV (pi )

Theorem (LeBruyn-Procesi 1990)

if char(K ) = 0, then I(Q, α) is generated by invariants of the formV 7→ Tr(V (p)) with p a cyclic path

so if Q has no oriented cycles, then I(Q, α) = K

Theorem (Donkin 1994)

I(Q, α) is generated by the coefficients of the characteristicpolynomial of all V (p) with p a cyclic path




i=1 λipi ) :=∑r

i=1 λiV (pi )









i=1 λipi ) :=∑r

i=1 λiV (pi )







Semi-Invariants

Assume K is infiniteFor σ ∈ ZQ0 we define a multiplicative character χσ : GLα → K ?

by

(A(x), x ∈ Q0) 7→∏x∈Q0

A(x)σ(x)

Definition

The space of semi-invariants of weight σ is

SI(Q, α)σ = {f ∈ K [Repα(Q)] | ∀A ∈ GLα A · f = χσ(A)}

The ring of semi-invariants is

SI(Q, α) = K [Repα(Q)]SLα =⊕σ

SI(Q, α)σ


Semi-Invariants


by

(A(x), x ∈ Q0) 7→∏x∈Q0

A(x)σ(x)

Definition





SI(Q, α)σ


Semi-Invariants


by

(A(x), x ∈ Q0) 7→∏x∈Q0

A(x)σ(x)

Definition





SI(Q, α)σ


GIT for quivers

Definition

a representation V is σ-(semi)stable if (V , 1) ∈ Repα(Q)⊕ χσ isGLα-(semi)stable

Repα(Q)ssσ (resp. Repα(Q)s) set of σ-semi-stable (resp. σ-stable)points

Theorem (King 1994)

(a) V is σ-semistable if and only if σ(α) = 0 and σ(dimW ) ≤ 0for every subrepresentation W of V

(b) V is σ-stable if and only if σ(α) = 0 and σ(dimW ) < 0 forevery proper subrepresentation W of V

(c) π : Repα(Q)ssσ → Proj(⊕

n SI(Q, α)nσ) is “good” quotient

(d) restriction to Repα(Q)s is geometric quotient


GIT for quivers

Definition



Theorem (King 1994)







GIT for quivers

Definition



Theorem (King 1994)







GIT for quivers

Definition



Theorem (King 1994)







GIT for quivers

Definition



Theorem (King 1994)







Schofield Semi-Invariants

Definition (Euler/Ringel Form)

for α, β dimension vectors

〈α, β〉 =∑x∈Q0

α(x)β(x)−∑a∈Q1

α(ta)β(ha)

Definition

if V ∈ Repα(Q), W ∈ Repβ(Q), define

dVW :

⊕x∈Q0

HomK (V (x),W (x))→⊕a∈Q1

HomK (V (ta),W (ha))

by(φ(x), x ∈ Q0) 7→ (φ(ha)V (a)−W (a)φ(ta), a ∈ Q1)



Definition (Euler/Ringel Form)

for α, β dimension vectors

〈α, β〉 =∑x∈Q0

α(x)β(x)−∑a∈Q1

α(ta)β(ha)

Definition

if V ∈ Repα(Q), W ∈ Repβ(Q), define

dVW :

⊕x∈Q0

HomK (V (x),W (x))→⊕a∈Q1

HomK (V (ta),W (ha))

by(φ(x), x ∈ Q0) 7→ (φ(ha)V (a)−W (a)φ(ta), a ∈ Q1)



suppose that 〈α, β〉 = 0dVW is a square matrix

Definition (Schofield 1991)

c(V ,W ) = cV (W ) = cV (W ) = det dVW

ker dVW = HomQ(V ,W ), coker dV

W = Ext1Q(V ,W ), so

c(V ,W ) = 0⇔ HomQ(V ,W ) = 0⇔ Ext1Q(V ,W ) = 0

Theorem (Schofield 1991)

cV ∈ SI(Q, β)〈α,·〉cW ∈ SI(Q, α)−〈·,β〉










cV ∈ SI(Q, β)〈α,·〉cW ∈ SI(Q, α)−〈·,β〉










cV ∈ SI(Q, β)〈α,·〉cW ∈ SI(Q, α)−〈·,β〉



Theorem (D.-Weyman 2000)

SI(Q, β) spanned by Schofield semi-invariants cV whereV ∈ Repα(Q) and α a dimension vector with 〈α, β〉 = 0

(a similar statement is true for cW ’s)


SI(Q, β)〈α,·〉 and SI(Q, α)−〈·,β〉 are dual and have same dimension
















Saturation for Semi-Invariants, etc.

char(K ) = 0


dim SI(Q, α)nσ is a polynomial in n

Theorem (D.-Weyman 2000, Generalized SaturationConjecture)

if dim SI(Q, α)σ = 0 then dim SI(Q, α)nσ = 0 for all n ≥ 1

Theorem (D.-Weyman 2011, Generalized Fulton Conjecture)


Theorem (Sherman 2016, gen. King-Tollu-Toumazet conj.)

if dim SI(Q, α)σ = 2 then dim SI(Q, α)nσ = n + 1 for all n ≥ 1



char(K ) = 0











char(K ) = 0











char(K ) = 0










Application to Littlewood-Richardson Coefficients

irreducible representations of GLp are Vλ where λ is a partition(Young diagram)

cνλ,µ = dim Hom(Vµ,Vλ ⊗ Vµ)GLp

is multiplicity of Vν in Vλ ⊗ Vµ.

Let Q = Tp,p,p:

◦ // · · · // ◦

��◦ // · · · // ◦ // ◦

◦ // · · · // ◦

??

Then cνλ,µ = dim SI(Q, α)σ for some α, σ anddim SI(Q, α)nσ = cνnλ,nµ


Application to Littlewood-Richardson Coefficients

irreducible representations of GLp are Vλ where λ is a partition(Young diagram)

cνλ,µ = dim Hom(Vµ,Vλ ⊗ Vµ)GLp

is multiplicity of Vν in Vλ ⊗ Vµ. Let Q = Tp,p,p:

◦ // · · · // ◦

��◦ // · · · // ◦ // ◦

◦ // · · · // ◦

??

Then cνλ,µ = dim SI(Q, α)σ for some α, σ anddim SI(Q, α)nσ = cνnλ,nµ


Saturation for LR-Coefficients, etc.


cνnλ,nµ is a polynomial in n

Theorem (Knutson-Tao 1999, Klyachko Saturation Conjecture)

if cνλ,µ = 0 then cnνnλ,nµ = 0 for all n ≥ 1

Theorem (Knutson-Tao-Woodward 2004, Fulton Conjecture)


Theorem (Sherman 2015, King-Tollu-Toumazet conj.)

if cνλ,µ = 2 then cnνnλ,nµ = n + 1 for all n ≥ 1
































Semi-Invariants as determinants

suppose that x1, x2, . . . , xr , y1, y2, . . . , ys ∈ Q0 (possible repetition)pi ,j linear combination of paths from xi to yj , andα is a dimension vector with

∑ri=1 α(xi ) =

∑sj=1 α(yj), then

V ∈ Repα(Q) 7→ det

V (p1,1) · · · V (ps,r )...

...V (ps,1) · · · V (ps,r )

is a semi-invariant of weight σ =

∑ri=1 1xi −

∑sj=1 1yj

Theorem (Domokos-Zubkov 2001)

SI(Q, α) is spanned by such semi-invariants


Semi-Invariants as determinants

suppose that x1, x2, . . . , xr , y1, y2, . . . , ys ∈ Q0 (possible repetition)pi ,j linear combination of paths from xi to yj , andα is a dimension vector with

∑ri=1 α(xi ) =

∑sj=1 α(yj), then

V ∈ Repα(Q) 7→ det

V (p1,1) · · · V (ps,r )...

...V (ps,1) · · · V (ps,r )

is a semi-invariant of weight σ =

∑ri=1 1xi −

∑sj=1 1yj

Theorem (Domokos-Zubkov 2001)

SI(Q, α) is spanned by such semi-invariants


Semi-Invariants of Generalized Kronecker Quiver

Q quiver with two vertices, x1, x2, and m arrows from x1 to x2α = (n, n) and σ = (1,−1), then:

Repα(Q) = Matmn,nSL(α) = SLn×SLn acts by left-right multiplication

Definition

for T = (T1,T2, . . . ,Tm) ∈ Matmd ,d define

fT (A1, . . . ,Am) = det(∑m

i=1 Ai ⊗ Ti )

semi-invariant of weight (d ,−d) and degree dn

(⊗ is Kronecker product for matrices)

Domokos-Zubkov Thm: fT ’s span SI(Q, α) = K [Matmn,n]SLn ×SLn





Definition


fT (A1, . . . ,Am) = det(∑m

i=1 Ai ⊗ Ti )



Domokos-Zubkov Thm: fT ’s span SI(Q, α) = K [Matmn,n]SLn ×SLn





Definition


fT (A1, . . . ,Am) = det(∑m

i=1 Ai ⊗ Ti )



Domokos-Zubkov Thm: fT ’s span SI(Q, α) = K [Matmn,n]SLn × SLn


Degree Bounds for Matrix Invariants

Theorem (Ivanyos-Qiao-Subrahmanyan)

K [Matmn,n]SLn × SLn generated by invariants of degree O(n816n2)

using general bounds for Nullcone [D. 2001]

Theorem (vanyos-Qiao-Subrahmanyan)

K [Matmn,n]SLn × SLn generated by invariants of degreeO(n4(n + 1)!2)

Theorem (D.-Makam 2017, Visu Makam thesis 2018)

K [Matmn,n]SLn × SLn generated by invariants of degree < mn4

(we may replace mn4 by n6)






















About the Proof ...

King’s criterion: A = (A1,A2, . . . ,Am) ∈ Matmn,n lies on thenullcone N (i.e., is not σ-semistable) if and only if there exists asubspaces W1,W2 of Kn such that

Ai (W1) ⊆W2

for all i , and dimW2 < dimW1

Theorem (D.-Makam 2017, uses IQS regularity lemma)

A = (A1, . . . ,An) ∈ N if and only if fT (A) = 0 for allT ∈ Matmn−1,n−1

fT has degree (n − 1)n, so together with Popov’s bound (fromearlier) we see that the invariant ring is generated in degree≤ dim Matmn,n((n − 1)n) < mn4


About the Proof ...


Ai (W1) ⊆W2






About the Proof ...


Ai (W1) ⊆W2






Ivanyos, Qiao Subrahmanyan Regularity Lemma

Theorem (Ivanyos, Qiao, Subrahmanyan 2016)

For given A1, . . . ,Am ∈ Matmn,n and generic T1, . . . ,Tm ∈ Matmd ,dthe rank of

∑mi=1 Ai ⊗ Ti is divisible by d .


Example

Example: n = m = 3 and take:

A1 =

0 1 0−1 0 00 0 0

,A2 =

0 0 10 0 0−1 0 0

,A3 =

0 0 00 0 10 −1 0

If T = (t1, t2, t3) ∈ Mat31,1, then

fT (A) = det

0 t1 t2−t1 0 t3−t2 −t3 0

= 0

because matrix is skew-symmetric of odd size


Example

Example: n = m = 3 and take:

A1 =

0 1 0−1 0 00 0 0

,A2 =

0 0 10 0 0−1 0 0

,A3 =

0 0 00 0 10 −1 0

If T = (t1, t2, t3) ∈ Mat31,1, then

fT (A) = det

0 t1 t2−t1 0 t3−t2 −t3 0

= 0

because matrix is skew-symmetric of odd size


However, if T = (T1,T2,T3) then

T1 ⊗ A1 + T2 ⊗ A2 + T3 ⊗ A3 =

0 T1 T2

−T1 0 T3

−T2 −T3 0

can be invertible, for example if we take T1 = I , then 0 I T2

−I 0 T3

−T2 −T3 0

∼ 0 I T2

−I 0 T3

−0 0 T3T2 − T2T3

now take

T2 =

(0 10 0

)and T3 =

(0 01 0

)so A = (A1,A2,A3) is semistable


Theorem (D.-Makam)

There exists A = (A1,A2, . . . ,Ad+1) ∈ Matd+1d2−1,d2−1 with

(a) for all e ≤ d and all T ∈ Matd+1e,e fT (A) = 0

(b) for some T ∈ Matd+1d ,d fT (A) 6= 0

For d = 3, take A = (A1,A2,A3,A4) ∈ Mat48,8 such that

4∑i=1

Ai ⊗ Ti =

T1 T3

−T2 T1 T3

−T2 T3

T1 T4

−T2 T1 T4

−T2 T4

T1 T2

−T2 T2 T1


Theorem (D.-Makam)

There exists A = (A1,A2, . . . ,Ad+1) ∈ Matd+1d2−1,d2−1 with

(a) for all e ≤ d and all T ∈ Matd+1e,e fT (A) = 0

(b) for some T ∈ Matd+1d ,d fT (A) 6= 0

For d = 3, take A = (A1,A2,A3,A4) ∈ Mat48,8 such that

4∑i=1

Ai ⊗ Ti =

T1 T3

−T2 T1 T3

−T2 T3

T1 T4

−T2 T1 T4

−T2 T4

T1 T2

−T2 T2 T1


Commutative Rank

K infinite fieldif A = (A1, . . . ,Am) ∈ Matmn,n then A(t) = t1A1 + · · ·+ tmAm

called linear matrixA(t) is n × n matrix whose entries are linear in t1, . . . , tm

Definition

commutative rank of A(t) is

cr(A(t)) = max{rankA(t1, . . . , tm) | t1, . . . , tm ∈ K}


Non-Commutative Rank

S = K (< t1, . . . , tm>) is free skew-field generated by t1, . . . , tmmay view A(t) as matrix with entries in K (< t1, . . . , tm>)

Definition

The noncommutative rank ncrk(A(t)) of A(t) is r , where r is therank of the image of A(t) : Sn → Sn as a free S-module

If T = (T1, . . . ,Tm) ∈ Matmd ,d then A(T ) =∑m

i=1 Ai ⊗ Ti isobtained by replacing ti by matrix Ti in A(t)

Theorem

ncrkA(t) = max{rank(A(T ))/d | d ≥ 1,T ∈ Matmd ,d}.

(this is an integer by regularity lemma)




Definition




Theorem






Definition




Theorem






Definition




Theorem




Inequalities for Comm. and Non-Comm. Rank

Clearly, crk(A(t)) ≤ ncrk(A(t)) We have seen that

A(t) =

0 t1 t2−t1 0 t3−t2 −t3 0

has commutative rank 2 and non-commutative rank 3

How much can discrepancy between commutative andnon-commutative rank be?

Theorem (Flanders)

ncrk(A(t)) ≤ 2 crk(A(t)).




A(t) =

0 t1 t2−t1 0 t3−t2 −t3 0



Theorem (Flanders)





A(t) =

0 t1 t2−t1 0 t3−t2 −t3 0



Theorem (Flanders)



Proof of Flander’s Theorem

Let r = crk(A(t))

For some s ∈ Km, r = rankA(s)without loss of generality we may assume that

A(s) =

(Ir 0

0 0

).

Then we have

A(t) =

(B(t) C (t)

D(t) 0

).

ncrk(A(t)) ≤ ncrk(B(t) C (t)) + ncrk(D(t)) ≤ r + r = 2r .

(B(t) and C (t) have r rows, D(t) has r columns)



Let r = crk(A(t))For some s ∈ Km, r = rankA(s)

without loss of generality we may assume that

A(s) =

(Ir 0

0 0

).

Then we have

A(t) =

(B(t) C (t)

D(t) 0

).





Let r = crk(A(t))For some s ∈ Km, r = rankA(s)without loss of generality we may assume that

A(s) =

(Ir 0

0 0

).

Then we have

A(t) =

(B(t) C (t)

D(t) 0

).






A(s) =

(Ir 0

0 0

).

Then we have

A(t) =

(B(t) C (t)

D(t) 0

).






A(s) =

(Ir 0

0 0

).

Then we have

A(t) =

(B(t) C (t)

D(t) 0

).




Theorem (D.-Makam 2016)

ncrk(A(t)) < 2 crk(A(t)).

(strict inequality)

this inequality is sharp, i.e., we can get arbitrarily close:

Let V be avector space of dimension 2p + 1 and for t ∈ V , defineA(t) :

∧p V →∧p+1 V by

A(t)(u) = u ∧ t

then crk(A(t)) =(2pp

)and ncrk(A(t)) =

(2p+1p

)and

ncrk(A(t))/ crk(A(t)) = 2p+1p+1 → 2 as p →∞




(strict inequality)

this inequality is sharp, i.e., we can get arbitrarily close:Let V be avector space of dimension 2p + 1 and for t ∈ V , defineA(t) :


A(t)(u) = u ∧ t


)and ncrk(A(t)) =

(2p+1p

)and





(strict inequality)

this inequality is sharp, i.e., we can get arbitrarily close:Let V be avector space of dimension 2p + 1 and for t ∈ V , defineA(t) :


A(t)(u) = u ∧ t


)and ncrk(A(t)) =

(2p+1p

)and



Documents

Invariant Theory for Quiverssites.lsa.umich.edu/.../2018/09/AuslanderSlides.pdf · Invariant Theory for Quivers Harm Derksen University of Michigan ... G homogeneous of degree d,