.

.

Newton-Okounkov polytopes of flag varieties
from cluster algebras
Naoki Fujita
The University of Tokyo

Joint work with Hironori Oya
Toric Topology 2019 in Okayama
November 20, 2019

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 20191 / 19Newton-Okounkov bodies and cluster algebras

1.

Newton-Okounkov bodies and cluster algebras2.

g-vectors as higher rank valuations

3.

Case of flag varieties

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 20192 / 19Newton-Okounkov bodies and cluster algebras

Toric degenerations
Let Z be an irreducible normal projective variety over C, and L an ample
line bundle on Z.
.
Definition.
A toric degeneration of (Z, L) is a flat morphism
π : X = Proj(R) → C
such that (π −1 (t), OX (1)|π−1 (t) ) ≃ (Z, L) for all t ∈ C× , and π −1 (0) is anirreducible
normal projective toric variety.
.
.
Aim.
to understand relations between the following two constructions of toricdegenerations:
Newton-Okounkov bodies (Anderson 2013),.

cluster algebras (Gross-Hacking-Keel-Kontsevich 2018).

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 20193 / 19Newton-Okounkov bodies and cluster algebras

Newton-Okounkov bodies
Z : an irreducible normal projective variety over C (m := dimC (Z)),
L : an ample line bundle on Z,τ ∈ H 0 (Z, L) : a nonzero section.
Assume that Z is rational, and fix an identification
C(Z) ≃ C(t1 , . . . , tm ).
We take a total order < on Zm , respecting the addition, which induces atotal order < on the set of Laurent monomials in t1 , . . . , tm . Then, thelowest term valuation v : C(Z) \ {0} → Zm is defined as follows:v(f /g) := v(f ) − v(g), andv(f ) := (a1 , . . . , am ) ⇔ f = cta11 · · · tamm + (higher terms w.r.t. <)for f, g ∈ C[t1 , . . . , tm ] \ {0}, where c ∈ C× .
Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 20194 / 19Newton-Okounkov bodies and cluster algebras

Newton-Okounkov bodies
We define a semigroup S(Z, L, v, τ ) ⊂ Z>0 × Zm , a real closed convex
cone C(Z, L, v, τ ) ⊂ R≥0 × Rm and a convex set ∆(Z, L, v, τ ) ⊂ Rm by
S(Z, L, v, τ ) := {(k, v(σ/τ k )) | k ∈ Z>0 , σ ∈ H 0 (Z, L⊗k ) \ {0}},C(Z, L, v, τ ) : the smallest real closed cone containing S(Z, L, v, τ ),∆(Z, L, v, τ ) := {a ∈ Rm | (1, a) ∈ C(Z, L, v, τ )}.
.

Definition (Lazarsfeld-Mustata 2009, Kaveh-Khovanskii 2012).
Theconvex set ∆(Z, L, v, τ ) is called a Newton-Okounkov body of Z.
.
.

Theorem (Anderson 2013).
If the semigroup S(Z, L, v, τ ) is finitely generated and saturated, thenthere exists a toric degeneration of (Z, L) to the normal projective toric.variety corresponding to ∆(Z, L, v, τ ).
Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 20195 / 19Newton-Okounkov bodies and cluster algebras

Newton-Okounkov bodies of flag varieties
G : a connected, simply-connected semisimple algebraic group over C,
B ⊂ G : a Borel subgroup,P+ : the set of dominant integral weights.
.
Definition.
The
quotient variety G/B is called the full flag variety.
.
.
Proposition.
There exists a natural bijective map∼

P+ −→ {line bundles on G/B generated by global sections},.

λ 7→ Lλ .

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 20196 / 19Newton-Okounkov bodies and cluster algebras

Newton-Okounkov bodies of flag varieties
The Newton-Okounkov bodies of (G/B, Lλ ) realize the following
representation-theoretic polytopes:
Berenstein-Littelmann-Zelevinsky’s string polytopes(Kaveh 2015),
Nakashima-Zelevinsky polytopes(F.-Naito 2017),
Feigin-Fourier-Littelmann-Vinberg polytopes(Feigin-Fourier-Littelmann 2017, Kiritchenko 2017).
.
Question.
How
are these polytopes related?
.

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 20197 / 19Newton-Okounkov bodies and cluster algebras

Newton-Okounkov bodies of flag varieties
The Newton-Okounkov bodies of (G/B, Lλ ) realize the following
representation-theoretic polytopes:
Berenstein-Littelmann-Zelevinsky’s string polytopes(Kaveh 2015),
Nakashima-Zelevinsky polytopes(F.-Naito 2017),
Feigin-Fourier-Littelmann-Vinberg polytopes(Feigin-Fourier-Littelmann 2017, Kiritchenko 2017).
.
Question.
How
are these polytopes related?
.

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 20197 / 19Newton-Okounkov bodies and cluster algebras

Cluster varieties
Let
A=

∪t

At =

∪Spec(C[x1 (t)±1 , . . . , xn (t)±1 , xn+1 (t)±1 , . . . , xm (t)±1 ])|{z} |{z}tunfrozen

frozenbe a cluster variety (Fock-Goncharov 2009), where t runs over the set ofseeds which are mutually mutation equivalent, and the tori are glued via
the following birational cluster mutations:(i ̸= k),xi (t)
(t)
(t)
∏∏
∗
′b−bµk (xi (t )) = xk (t)−1 (xj (t) k,j +xj (t) k,j ) (i = k)

(t)
(t)
bk,j >0bk,j <0

if t′ = µk (t), where B(t) = (bk,j )k,j is the exchange matrix of t.
.
(t)Definition (Berenstein-Fomin-Zelevinsky 2005).
Thering C[A] of regular functions is called an upper cluster algebra.
.
Naoki Fujita (The University of Tokyo)

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Nov. 20, 20198 / 19Newton-Okounkov bodies and cluster algebras

Cluster varieties
Denoting the dual torus of At by A∨t , we have∪∪“mirror′′
A=At ←−−−−−−−→ A∨ =A∨t.
tt

The space A∨ is called the Fock-Goncharov dual, and defined to be theLanglands dual of the X -cluster variety. Since the gluing maps of A∨ aregiven by subtraction-free rational functions, we obtain the set A∨ (RT ) of
RT -valued points, where RT is a semifield (R, max, +). More precisely,TmA∨ (RT ) is defined by gluing A∨t (R ) = R via the following tropicalizedcluster mutation:T∨T′′µTk : A∨t (R ) → At′ (R ), (g1 , . . . , gm ) 7→ (g1 , . . . , gm ),wheregi′ ={(t)(t)gi + max{bk,i , 0}gk − bk,i min{gk , 0}
−gk

Naoki Fujita (The University of Tokyo)(i ̸= k),(i = k).

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Nov. 20, 20199 / 19Newton-Okounkov bodies and cluster algebras

Gross-Hacking-Keel-Kontsevich’s toric degenerations
For each t, we haveTm
A∨ (RT ) ≃ A∨t (R ) = R .
Tm
We fix Ξ ⊂ A∨ (RT ) such that its image in A∨t (R ) = R , denoted by Ξt ,is an m-dimensional rational convex polytope for each t.
.

Theorem (Gross-Hacking-Keel-Kontsevich 2018).
Under some conditions on A and Ξ, there exists a Z≥0 -graded ring Rt foreach t together with a flat morphism πt : Proj(Rt ) → Cm such thatπt−1 (z) for z ∈ (C× )m is a normal projective variety which
compactifies A,.

πt−1 (0) is the normal projective toric variety corresponding toΞt ⊂ Rm .

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201910 / 19g-vectors as higher rank valuations

1.

Newton-Okounkov bodies and cluster algebras2.

g-vectors as higher rank valuations

3.

Case of flag varieties

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201911 / 19g-vectors as higher rank valuations

extended g-vectors.

Theorem (Fomin-Zelevinsky 2007, Derksen-Weyman-Zelevinsky 2010,
Gross-Hacking-Keel-Kontsevich 2018).
For all t, t′ and 1 ≤ i ≤ m,xi (t′ ) ∈ x1 (t)g1 · · · xm (t)gm (1 +∑Zŷ1a1 · · · ŷnan )0̸=(a1 ,...,an )∈Zn
≥0for some gt (xi (t′ )) := (g1 , . . . , gm ) ∈ Zm , where(t).

(t)

ŷi := x1 (t)bi,1 · · · xm (t)bi,m .

The vector gt (xi (t′ )) ∈ Zm is called the extended g-vector of xi (t′ ).

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201912 / 19g-vectors as higher rank valuations

g-vectors as higher rank valuations.

Definition (Qin 2017).
For each t, define a partial order ≤t on Zm by∑(t)(t)
g ′ ≤t g ⇔ g − g ′ ∈Z≥0 (bi,1 , . . . , bi,m ).
1≤i≤nThis≤t is called the dominance order associated with t.
.
Since At is an open subvariety of A, we haveC(A) = C(At ) = C(x1 (t), . . . , xm (t)).
We fix a total order ≤t on Zm which refines the dominance order ≤t .
.
Definition.
For each t, define a valuation vt : C(A) \ {0} → Zm to be the lowest term.valuation associated with ≤t .
Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201913 / 19g-vectors as higher rank valuations

g-vectors as higher rank valuations.
Proposition.
′′′
For. all t, t and 1 ≤ i ≤ m, the equality vt (xi (t )) = gt (xi (t )) holds.
.
Proposition.
The valuation vt generalizes the extended g-vectors of the theta functionbasis.
.
.
Corollary.
Under Gross-Hacking-Keel-Kontsevich’s toric degeneration, the polytope−1Ξ
. t is identical to the Newton-Okounkov body of πt (z) associated with vt .
Let Z be an irreducible normal projective variety which is birational to A.
.
Problem.
ComputeNewton-Okounkov bodies of Z associated with vt .
.
Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201914 / 19

Case of flag varieties

1.

Newton-Okounkov bodies and cluster algebras2.

g-vectors as higher rank valuations

3.

Case of flag varieties

Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201915 / 19

Case of flag varieties

Cluster structures
Let U − be the unipotent radical of the opposite Borel subgroup B − , and⊔
G/B =BwB/Bw∈W

the Bruhat decomposition of G/B, where W is the Weyl group.
.
Definition.
For w ∈ W , the unipotent cell Uw− ⊂ G is defined by.

Uw− := BwB ∩ U − .

.

Theorem (Berenstein-Fomin-Zelevinsky 2005).
The
coordinate ring C[Uw− ] admits an upper cluster algebra structure.
.
There exists w0 ∈ W , called the longest element, such that the naturalprojection G ↠ G/B induces an open embedding Uw−0 ,→ G/B.
Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201916 / 19

Case of flag varieties

Relation with string polytopes
Each reduced word i = (i1 , . . . , im ) for w0 induces a seed ti for Uw−0 .
.
Example.
Let G = SL4 (C), and i = (1, 2, 1, 3, 2, 1). Then, the exchange matrix(t )B(ti ) = (bi,ji )i,j corresponds to the following quiver:
72
7◦o72

' o◦o◦
2(ti )(ti )Forinstance,b=1andb=−1(ifdefinedand)
ifthere exists i → j.
i,jj,i.

.

Theorem (F.-Oya).
The Newton-Okounkov body ∆(G/B, Lλ , vti , τ ) is unimodularly
equivalentto the string polytope ∆i (λ) associated with i and λ.
.
Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201917 / 19

Case of flag varieties

Relation with Nakashima-Zelevinsky polytopes
Berenstein-Fomin-Zelevinsky (1996) and Berenstein-Zelevinsky (1997)∼
introduced the twist automorphism ηw : Uw− −→ Uw− to solve afactorization problem, called the Chamber Ansatz. It is defined by
ηw (u) := [uT w]− ,where uT is the transpose of u, and w ∈ G is a lift for w. In addition,[uT w]− is the U − -part of uT w under the Gaussian decomposition. Then,→
there exists a sequence −µ = µk1 µk2 · · · µkl of mutations whichcorresponds to ηw in some sense (Geiss-Leclerc-Schröer 2011, 2012, . . . ).
We define a seed t̂i by→t̂i := −µ −1 (ti ).
.

Theorem (F.-Oya).
The Newton-Okounkov body ∆(G/B, Lλ , vt̂i , τ ) is unimodularly equivalenteto. the Nakashima-Zelevinsky polytope ∆i (λ) associated with i and λ.
Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201918 / 19

Case of flag varieties

Simply-laced caseIn simply-laced case, using the monoidal categorification of the cluster
algebra by Kang-Kashiwara-Kim-Oh (2018), we obtain the following.
.

Theorem (F.-Oya).
If G is of simply-laced, then the following hold.
(1) For each t, the Newton-Okounkov body ∆(G/B, Lλ , vt , τ ) is arational convex polytope which gives a toric degeneration of G/B.
(2) If t′ = µk (t), then the following equality holds:.

∆(G/B, Lλ , vt′ , τ ) = µTk (∆(G/B, Lλ , vt , τ )).

.
Corollary.
If G is of simply-laced, then the string polytopes ∆i (λ) and thee i (λ) are all related by tropicalized
Nakashima-Zelevinsky polytopes ∆.cluster mutations.
Naoki Fujita (The University of Tokyo)

NO polytopes from cluster algebras

Nov. 20, 201919 / 19