gröbner theory of grassmannian cluster algebras

Grobner theory of Grassmannian cluster algebras

Lara Bossinger (jt. Fatemeh Mohammadi and Alfredo Najera Chavez)

Universidad Nacional Autonoma de Mexico, IM-Oaxaca

Interdisciplinary applications of cluster algebras

SIGMA 17 (2021), 59 Lara Bossinger (jt. Fatemeh Mohammadi and Alfredo Najera Chavez) 0/ 11

Motivation

Idea: Want to relate a cluster algebra A with the Grobner theory of anideal I s.t. A = k[x1, . . . , xm]/I .

Theorem (Fomin–Williams–Zelevinsky 20)

Let a be a finite type cluster algebra with XA the set of all clustervariables. Let IA ⊂ k[XA] be the saturation of the ideal generated by allexchange relations. Then

A ∼= k[XA]/IA.

Motivation

Idea: Want to relate a cluster algebra A with the Grobner theory of anideal I s.t. A = k[x1, . . . , xm]/I .

Theorem (Fomin–Williams–Zelevinsky 20)

Let a be a finite type cluster algebra with XA the set of all clustervariables. Let IA ⊂ k[XA] be the saturation of the ideal generated by allexchange relations. Then

A ∼= k[XA]/IA.

Grassmannian cluster algebraLet Ak,n be the homogeneous coordinate ring of Gr(k , n) under its Pluckerembedding

Gr(k,n) ↪→ P(nk)−1, M 7→ [det(MI )]I∈([n]

[Scott 06] Ak,n is a cluster algebra and of finite type if and only if(k , n) ∈ {(2, n), (3, 6), (3, 7), (3, 8)}.

Example

1 For A2,n all cluster variables are Plucker coordinates pij and the idealIA2,n = I2,n is the Plucker ideal generated by

pijpkl − pikpjl + pilpjk

2 For A3,6 all Plucker coordinates pijk are cluster variables and there aretwo more: x , y that are quadratic binomials in pijk . So I3,6 containsthe Plucker ideal.

Example

A minimal generating set of I3,6 ⊂ C[p123, . . . , p456, x , y ]:

Grobner fan and (totally positive) tropicalization

Definition (Mora–Robbiano 88)

For a homogeneous ideal J ⊂ C[x1, . . . , xn] its Grobner fan GF (J) is Rn

with fan structure defined by

v ,w ∈ C ◦ ⇔ inv (J) = inw (J).

Notation: inC (J) := inw (J),w ∈ C ◦ and AC := C[x1, . . . , xn]/inw (J)

Definition (Speyer–Sturmfels 04, Speyer–Williams 05)

There is a distinguished subfan of GF (J) called the tropicalization of J:

Trop(J) := {w ∈ Rn : inw (J) 63 monomials}.

If J ⊂ R[x1, . . . , xn] then the totally positive part Trop+(J) ⊂ Trop(J) isthe closed subfan:

Trop+(J) := {w ∈ Trop(J) : inw (J) ∩ R+[x1, . . . , xn] = ∅}.

v ,w ∈ C ◦ ⇔ inv (J) = inw (J).

Trop+(J) := {w ∈ Trop(J) : inw (J) ∩ R+[x1, . . . , xn] = ∅}.SIGMA 17 (2021), 59 Lara Bossinger (jt. Fatemeh Mohammadi and Alfredo Najera Chavez) 4/ 11

Example

Take J = (x21x

22 + x4

1 − x2x33 ) ⊂ C[x1, x2, x3]. Then GF (J) is R3 with the

fan structure and Trop(J) its one-skeleton, Trop+(J) is spanned by r1, r2:

w3 = 0

×L = 〈(1, 1, 1)〉

(x41 )

(x2x33 )

(x21 x

22 + x4

1 )(x2

1 x22 − x2x

(x41 − x2x

Note: B〈r1,r2〉 = {xa : x2x33 6 | xa} gives standard monomial basis for A.

Example

Take J = (x21x

22 + x4

w3 = 0

×L = 〈(1, 1, 1)〉

(x41 )

(x2x33 )

(x21 x

22 + x4

1 )(x2

1 x22 − x2x

(x41 − x2x

Example

Take J = (x21x

22 + x4

w3 = 0

×L = 〈(1, 1, 1)〉

(x41 )

(x2x33 )

(x21 x

22 + x4

1 )(x2

1 x22 − x2x

(x41 − x2x

Example

Take J = (x21x

22 + x4

w3 = 0

×L = 〈(1, 1, 1)〉

(x41 )

(x2x33 )

(x21 x

22 + x4

1 )(x2

1 x22 − x2x

(x41 − x2x

Family of idealsLet C ∈ GF (J) be a maximal cone and choose r1, . . . , rm representativesof primitive ray generators of C ∈ GF(J)/L. Let r be the matrix with rowsr1, . . . , rm.

Define for f =∑

α∈Zn≥0

cαxα ∈ J

µ(f ) := ( mincα 6=0{r1 · α}, . . . , min

cα 6=0{rm · α}) ∈ Zm.

In C[t1, . . . , tm][x1, . . . , xn] the lift of f is

f := f (tr·e1x1, . . . , tr·enxn)t−µ(f ) =

∑α∈Zn

cαxαtr·α−µ(f ).

Theorem (B.–Mohammadi–Najera Chavez)

The lifted ideal J := (f : f ∈ J) ⊂ C[t1, . . . , tm][x1, . . . , xn] is generated by{g : g ∈ G}, where G is a Grobner basis for J and C .Moreover, A is a free C[t1, . . . , tm]-module with standard monomial basisBC .

Family of idealsLet C ∈ GF (J) be a maximal cone and choose r1, . . . , rm representativesof primitive ray generators of C ∈ GF(J)/L. Let r be the matrix with rowsr1, . . . , rm. Define for f =

∑α∈Zn

≥0cαxα ∈ J

µ(f ) := ( mincα 6=0{r1 · α}, . . . , min

cα 6=0{rm · α}) ∈ Zm.

∑α∈Zn

≥0cαxα ∈ J

µ(f ) := ( mincα 6=0{r1 · α}, . . . , min

cα 6=0{rm · α}) ∈ Zm.

∑α∈Zn

≥0cαxα ∈ J

µ(f ) := ( mincα 6=0{r1 · α}, . . . , min

cα 6=0{rm · α}) ∈ Zm.

∑α∈Zn

Example

(x2x33 )

(x21 x

22 − x2x

(x41 − x2x

Take f = x21x

22 + x4

1 − x2x33 ∈ C[x1, x2, x3] and consider

in GF ((f )) the maximal cone C spanned by the rows ofr :=

(1 4 01 −2 0

)and L.

We compute

f (t1, t2) = f (t1t2x1, t41 t−22 x2, x3)t−4

= t61x

22 + t6

2x41 − x2x

f (0, 0) = −x2x33 = inC (f ),

f (0, 1) = x41 − x2x

33 = inr1(f ),

f (1, 0) = x21x

22 − x2x

33 = inr2(f ),

f (1, 1) = f .

So A = C[t1, t2][x1, x2, x3]/(t61x

22 + t6

2x41 − x2x

Example

(x2x33 )

(x21 x

22 − x2x

(x41 − x2x

Take f = x21x

22 + x4

(1 4 01 −2 0

)and L. We compute

f (t1, t2) = f (t1t2x1, t41 t−22 x2, x3)t−4

= t61x

22 + t6

2x41 − x2x

f (0, 0) = −x2x33 = inC (f ),

f (0, 1) = x41 − x2x

33 = inr1(f ),

f (1, 0) = x21x

22 − x2x

33 = inr2(f ),

f (1, 1) = f .

So A = C[t1, t2][x1, x2, x3]/(t61x

22 + t6

2x41 − x2x

Example

L×(x2x

(x21 x

22 − x2x

(x41 − x2x

Take f = x21x

22 + x4

(1 4 01 −2 0

)and L. We compute

f (t1, t2) = f (t1t2x1, t41 t−22 x2, x3)t−4

= t61x

22 + t6

2x41 − x2x

f (0, 0) = −x2x33 = inC (f ),

f (0, 1) = x41 − x2x

33 = inr1(f ),

f (1, 0) = x21x

22 − x2x

33 = inr2(f ),

f (1, 1) = f .

So A = C[t1, t2][x1, x2, x3]/(t61x

22 + t6

2x41 − x2x

Example

L×(x2x

(x21 x

22 − x2x

(x41 − x2x

Take f = x21x

22 + x4

(1 4 01 −2 0

)and L. We compute

f (t1, t2) = f (t1t2x1, t41 t−22 x2, x3)t−4

= t61x

22 + t6

2x41 − x2x

f (0, 0) = −x2x33 = inC (f ),

f (0, 1) = x41 − x2x

33 = inr1(f ),

f (1, 0) = x21x

22 − x2x

33 = inr2(f ),

f (1, 1) = f .

So A = C[t1, t2][x1, x2, x3]/(t61x

22 + t6

2x41 − x2x

Example

L×(x2x

(x21 x

22 − x2x

(x41 − x2x

Take f = x21x

22 + x4

(1 4 01 −2 0

)and L. We compute

f (t1, t2) = f (t1t2x1, t41 t−22 x2, x3)t−4

= t61x

22 + t6

2x41 − x2x

f (0, 0) = −x2x33 = inC (f ),

f (0, 1) = x41 − x2x

33 = inr1(f ),

f (1, 0) = x21x

22 − x2x

33 = inr2(f ),

f (1, 1) = f .

So A = C[t1, t2][x1, x2, x3]/(t61x

22 + t6

2x41 − x2x

For (k, n) ∈ {(2, n), (3, 6)} there exists a unique maximal cone C in theGrobner fan of Ik,n such that

1 inC (Ik,n) is generated by products of non-compatible cluster variables(i.e. it’s the Stanley–Reisner ideal of the cluster complex);

2 canonically Ak,n∼= Auniv

k,n where universal coefficients1:1↔ rays of C ;

3 standard monomial basis BC = basis of cluster monomials;

4 C ∩ Trop(Ik,n) = Trop+(Ik,n) which is a geometric realization of thecluster complex:

rays of Trop+(Ik,n)1:1←→ cluster variables

max cones τs ∈ Trop+(Ik,n)1:1←→ seeds s of Ak,n

where the toric variety TV (∆(Ak,n, gs)) is Proj(Aτs ).

[Ilten–Najera Chavez–Treffinger]: generalized (1) for graded clusteralgebras of finte type and (2)/(3) for ADE types.

Arbitrary Grassmannians (in progress)[GHKK 18]/[Fujita–Oya 20]/[B.–Cheung–Magee–Najera Chavez]:

s seedof Ak,n

valuation

gs : Ak,n \ {0} → Zd toric degeneration

of Gr(k,n)

[B.21]/[Kaveh–Manon 19]:

toric degenerationsof Gr(k , n) induced

by valuations on Ak,n

←→maximal cones in

Trop(J) for some JAk,n

∼= k[x1, . . . , xm]/J

Conjecture (B.)

For every seed s of Ak,n exists a maximal prime cone τs in Trop+(J) for anappropriate ideal J with Ak,n

∼= k[x1, . . . , xm]/J, s.t. if J is appropriate fortwo adjacent seeds s, s ′ then τs and τs′ share a facet.

Remark: J is appropriate for s if im(gs) is generated by gs(x1), . . . , gs(xm).

s seedof Ak,n

valuation

of Gr(k,n)

∼= k[x1, . . . , xm]/J

Conjecture (B.)

s seedof Ak,n

valuation

of Gr(k,n)

∼= k[x1, . . . , xm]/J

Conjecture (B.)

∼= k[x1, . . . , xm]/J,

s.t. if J is appropriate fortwo adjacent seeds s, s ′ then τs and τs′ share a facet.

s seedof Ak,n

valuation

of Gr(k,n)

∼= k[x1, . . . , xm]/J

Conjecture (B.)

Examples1 For every seed s of A2,n

im(gs) = 〈gs(pij) : 1 ≤ i < j ≤ n〉,

so the Plucker ideal is appropriate for all seeds.

2 For A3,6 there exist seeds s for which gs(x) or gs(y) is not in〈gs(pijk) : 1 ≤ i < j < k ≤ 6〉,e.g. seeds from the plabic graphs

so the Plucker ideal is not appropriate for all seeds.

3 Conjecturally, the ideal IA of a finite type cluster algebra A isappropriate for all seeds.

im(gs) = 〈gs(pij) : 1 ≤ i < j ≤ n〉,

2 For A3,6 there exist seeds s for which gs(x) or gs(y) is not in〈gs(pijk) : 1 ≤ i < j < k ≤ 6〉,

e.g. seeds from the plabic graphs

im(gs) = 〈gs(pij) : 1 ≤ i < j ≤ n〉,

References

BMN Lara Bossinger, Fatemeh Mohammadi, Alfredo Najera Chavez. Families of Grobner Degenerations, Grassmannians andUniversal Cluster Algebras SIGMA 17 (2021), 59

Gr(3,6) Lara Bossinger. Grassmannians and universal coefficients for cluster algebras: computational data for Gr(3,6).https://www.matem.unam.mx/~lara/clusterGr36

B21 Lara Bossinger. Full-Rank Valuations and Toric Initial Ideals. Int. Math. Res. Not. rnaa071 (2021) 10

FO20 Naoki Fujita and Hironori Oya: Newton-Okounkov polytopes of Schubert varieties arising from cluster structures.arXiv:2002.09912

FZ07 Sergey Fomin and Andrei Zelevinsky. Cluster algebras. IV. Coefficients. Compos. Math. 143, no. 1, 112–164 (2007)

FWZ20 Sergey Fomin, Lauren Williams and Andrei Zelevinsky.Introduction to Cluster Algebras Chapter 6 arxiv:2008.09189

GHKK18 Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich. Canonical bases for cluster algebras. J. Amer. Math.Soc., 31(2):497–608 (2018)

INT21 Nathan Ilten, Alfredo Najera Chavez and Hipolito Treffinger. Deformation Theory for Finite Cluster Complexes.arXiv:2111.02566

KM19 Kiumars Kaveh and Christopher Manon. Khovanskii bases, higher rank valuations, and tropical geometry. SIAM J.Appl.Algebra Geom., 3(2):292–336 (2019)

MR88 Teo Mora and Lorenzo Robbiano. The Grobner fan of an ideal. Computational aspects of commutative algebra. J.Symbolic Comput. 6 (1988), no. 2-3, 183–208

Reading Nathan Reading. Universal geometric cluster algebras. Math. Z. 277(1-2):499–547 (2014)

Scott Joshua S. Scott. Grassmannians and cluster algebras. Proc. London Math. Soc. (3) 92 (2006), no. 2, 345–380.

SS04 David Speyer and Bernd Sturmfels. The tropical Grassmannian. Adv. Geom. 4 (2004), no. 3, 389–411.

SW05 David Speyer and Lauren Williams. The tropical totally positive Grassmannian. J. Algebraic Combin. 22 (2005), no. 2,189–210

gröbner theory of grassmannian cluster algebras

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