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QUANTUM FROBENIUS SPLITTINGS AND CLUSTER
STRUCTURES
JINFENG SONG
Abstract. We prove that the duals of the quantum Frobenius morphisms and
their splittings by Lusztig are compatible with quantum cluster monomials.
After specialization, we deduce that the canonical Frobenius splittings on flag
varieties are compatible with cluster algebra structures on Schubert cells.
1. Introduction
Let g=n−⊕h⊕nbe a triangular decomposition of a symmetrizable Kac-Moody
Lie algebra, and Uq(n) be the quantized universal enveloping algebra associated to
n, with the Lusztig integral form Uq(n) which is the Z[q±1]-subalgebra generated
by divided powers.
Let lbe an odd integer which is coprime to all the root lengths of g, and εbe
a primitive l-th root of unity. Lusztig [18, 19] constructed a quantum Frobenius
morphism Fr :Uε(n)!U1(n) from the quantized universal enveloping algebra
at l-th root of unity to the classical universal enveloping algebra and a splitting
Fr′:U1(n)!Uε(n) such that Fr ◦Fr′=id. These two maps have important
applications in representation theory and geometry. To name a few, the map Fr
plays essential role in the study of representations of reductive groups over positive
characteristic [1,14], and the map Fr′is used to construct Frobenius splittings of
Schubert varieties in [21].
Lusztig’s original definitions of these two maps are only derived through brute
force computations. It is desirable to obtain more conceptual understandings for
these maps. Several progresses have been made towards this question. McGerty
[23] gave a Hall algebra construction of the map Fr. Qi [25] categorified Fr for
g=sl2at prime roots of unity via p-DG algebras. The map Fr′seems to be
more mysterious, and no similar results are known. In the current paper we make
another attempt to answer this question from the cluster theory point of view.
Our approach treats both of the maps Fr and Fr′simultaneously.
Let Aq(n) be the graded dual of the integral form Uq(n). Take any element
win the Weyl group of g. Assume l(w) = r, where l(·) is the length function.
Following [17], the (integral) quantized coordinate ring Aq(n(w)) of the unipotent
subgroup N(w) = N∩w−1N−wis defined as a Z[q±1]-subalgebra of Aq(n), which
is spanned by a dual PBW-type basis associated to a reduced expression of w. It
follows from Kang–Kashiwara–Kim–Oh [15] and Goodearl–Yakimov [12] that the
Z[q±1/2]-algebra Aq1/2(n(w)) = Z[q±1/2]⊗Z[q±1]Aq(n(w)) has a quantum cluster
algebra structure of rank rin the sense of Berenstein–Zelevinsky [5]. For any
cluster xof Aq1/2(n(w)) and a∈Nr, we denote xato be the corresponding quantum
cluster monomial (see §2.1 for definitions).
2020 Mathematics Subject Classification. 13F60,17B37.
1
2 JINFENG SONG
The Z[ε]-algebras Aε(n) and A1(n) are defined to be the base change of Aq1/2(n)
via q1/27! ε(l+1)/2and q1/27! 1 respectively. They are naturally isomorphic to
the graded duals of Uε(n) and U1(n). By taking duals of the map Fr and Fr′, we
get Z[ε]-linear maps Fr∗:A1(n)!Aε(n) and Fr′,∗:Aε(n)!A1(n). For any
element fin Aq1/2(n), we use εf(resp., 1f) to denote its image in Aε(n) (resp.,
A1(n)). The main result of this paper is the following.
Theorem 1. Let wbe a Weyl group element of length rand xa(a∈Nr)be a
quantum cluster monomial in Aq1/2(n(w)). We have
Fr∗(1xa) = εxlaand Fr′,∗(εxa) = 1xa/l.
Here xa/l is understood as 0if a∈ lNr.
In their study of the connections between standard monomial basis and dual
canonical basis, Caldero and Littelmann [8] obtained part of Theorem 1 for certain
cluster monomials associated to reduced expressions. See Remark 11 for the precise
connection to their work.
Theorem 1 is proved in §2.8. It is noteworthy that our result is related to the
question on the compatibility between canonical basis and quantum Frobenius.
The canonical basis by Lusztig and Kashiwara is a particular linear basis of
Uq(n). In [23, Remark 5.10], McGerty raised a question on the compatibility
between maps Fr and Fr′with canonical basis. It is later shown by Baumann [2]
that in general they are not compatible in the strict sense, meaning that these
maps do not send a basis element to another basis element or zero, but they are
compatible up to certain filtrations.
The canonical basis of Uq(n) gives the dual canonical basis of its graded dual
Aq(n). By Kang–Kashiwara–Kim–Oh [15] and Qin [24], cluster monomials form
a subset of dual canonical basis up to scalar. Therefore we expect that our result
can provide new insights for McGerty’s question.
As an application of Theorem 1, we show the maps Fr∗and Fr′,∗preserve the
quantized coordinate ring of the unipotent subgroup associated to the same Weyl
group element (Corollary 12). To be more precise, for any win the Weyl group,
we obtain the following two inclusions,
Fr∗A1(n(w))⊂Aε(n(w)) and Fr′,∗Aε(n(w))⊂A1(n(w)).
Here Aε(n(w)) and A1(n(w)) are Z[ε]-subalgebras of Aε(n) and A1(n) respec-
tively, which are spanned by images of dual PBW-type basis under corresponding
base changes. The first inclusion also follows from the compatibility between Fr
and braid group actions, when gis of finite type. The second inclusion is more
subtle, since there is no obvious compatibility between Fr′and braid group actions.
We next discuss some geometric applications of our result. Assume gis of
finite type, and lis a prime number for the rest of this section. Let kbe an
algebraically closed field of characteristic l. Let Gbe the reductive group defined
over kassociated to the Lie algebra g,Bbe the standard Borel subgroup and
G/B be the associated flag variety. For elements wand vin the Weyl group of
G, we denote Cw=BwB/B to be the Schubert cell,Cv=B−vB/B to be the
opposite Schubert cell and Cv
w=Cw∩Cvto be the open Richardson variety.
Kumar–Littelmann [21] showed that the map Fr′,∗provides a quantum lift of a
Frobenius splitting of the flag variety G/B. By [21, Theorem 6.4] and [4, Theorem
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 3
4.1.15], this splitting is the unique B-canonical splitting of G/B and moreover the
splitting compatibly splits all the Schubert variety and opposite Schubert variety.
Therefore the canonical splitting of G/B induces canonical splittings of Schubert
cells as well as open Richardson varieties.
One can identify the unipotent subgroup N(w) with the Schubert cell Cw−1
via the map n7! n·w−1B/B. Under this isomorphism the k-algebra Ak(n(w))
obtained from Aq1/2(n(w)) by specializing q1/2at 1 is canonically isomorphic to
the coordinate ring of the Schubert cell Cw−1[17, Theorem 4.44]. One can show
that Ak(n(w)) carries a (classical) cluster algebra structure via specialization, by
applying similar arguments as in [10]. Our result implies the compatibility between
the canonical splittings on Schubert cells and their cluster algebra structures.
Corollary 2 (Proposition 14 & Corollary 17).
(1) The restriction of Fr′,∗provides a quantum lift of the canonical splitting of
the Schubert cell.
(2) The canonical splittings of Schubert cells are compatible with their cluster
algebra structures, that is, they divide the exponents by lwhen acting on
any cluster monomials.
We remark that Benito–Muller–Rajchgot–Smith showed in [3, Theorem 3.7]
that any upper cluster algebra (over k) admits a unique Frobenius splitting which
divides the exponents by lwhen acting on any cluster monomials. We call it the
cluster splitting. The associated upper cluster algebra of Ak(n(w)) is canonically
isomorphic to the coordinate ring of the open Richardson variety Ce
w−1. Then
another way to state Corollary 2 (2) is that the canonical splitting of the open
Richardson variety of the form Ce
w(w∈W) coincides with its cluster splitting.
The (upper) cluster algebra structures on general open Richardson varieties are
recently obtained by [7] and [11]. We expect that the same statement remains
true for any open Richardson varieties.
As another geometric application of Theorem 1, we show that the canonical
splitting of the Schubert cell is compatible with reduction maps.
Suppose w=v′vin the Weyl group of gsuch that l(w) = l(v′) + l(v). Following
[22, §4.3], define the reduction map between Schubert cells,
πw
v:Cw−1!Cv−1, b ·w−1B/B 7! b·v−1B/B.
Corollary 3. The canonical splittings on Schubert cells are compatible with re-
duction maps, that is, we have the following commuting diagram,
Ak(n(v)) Ak(n(w))
Ak(n(v)) Ak(n(w)).
(πw
v)∗
φvφw
(πw
v)∗
Here (πw
v)∗is the comorphism, and the map φv(resp., φw) is the canonical splitting
of the Schubert cell Cv−1(resp., Cw−1).
Corollary 3 is proved in §3.5.
The paper is organized as follows. In §2 we recall relevant results and prove
the main theorem. In §3 we recall the concept of Frobenius splittings and deduce
geometric consequences by specializing qat 1.
4 JINFENG SONG
Acknowledgment: The author would like to thank his supervisor Huanchen Bao
for many helpful discussions. The author is supported by Huanchen Bao’s MOE
grant A-0004586-00-00 and A-0004586-01-00. We thank the referee for valuable
suggestions.
2. Quantum Frobenius splittings
2.1. Quantum cluster algebras. In this subsection, we recall quantum cluster
algebras following [5] and [15, §5].
Fix a finite index set J=Jex ⊔Jfz with the decomposition into exchangeable
indices set Jex and frozen indices set Jfz. Let Fbe a skew field over Q(q1/2). Here
q1/2is an indeterminate.
Let Λ = (λij )i,j ∈Jbe a skew-symmetric integer valued J×J-matrix. By abuse
of notations, we will also denote by Λ the skew-symmetric bilinear form on ZJ
given by Λ(ei,ej) = λij. Here {ei|i∈J}is the standard basis of ZJ. A tuple
x= (xi)i∈Jconsisting of elements in Fis called Λ-commutative if xixj=qλij xjxi,
for i, j ∈J. For such a tuple xand a= (ai)i∈J∈ZJ, define the element
xa=q1/2Pi>j aiajλij Y
i∈J
xai
iin F. (1)
Here we take an arbitrary total order on J, and the product is taken increasingly.
One can check that this product is independent of the choice of the total order.
For any a,b∈ZJ, by direct computations one has
xaxb=q1/2Λ(a,b)xa+band xaxb=qΛ(a,b)xbxa.(2)
A Λ-commutative tuple x= (xi)i∈Jis called algebraically independent if ele-
ments xa(a∈ZJ) are linearly independent in F.
Let e
B= (bij)(i,j)∈J×Jex be an integer-valued J×Jex matrix. The pair (Λ,e
B) is
called compatible if for any j∈Jex and i∈J, we have
X
k∈J
bkj λki =δij dj,(3)
for some positive integer dj. This is equivalent to the requirement that the matrix
D=e
BTΛ consists of two blocks: the Jex ×Jex diagonal matrix with positive
diagonal entries dj(j∈Jex), and the Jex ×Jfz zero matrix. The matrix Dis called
the skew-symmetrizer of (Λ,e
B).
Aquantum seed S= (x,Λ,e
B) in Fconsists of a compatible pair (Λ,e
B) and
a Λ-commutative algebraically independent tuple x= (xi)i∈Jin F. The tuple
xis called the cluster of the quantum seed S, and its entries are called cluster
variables. Elements in {xi}i∈Jex are called exchangeable variables, while elements
in {xi}i∈Jfz are called frozen variables.
Let S= (x= (xi)i∈J,Λ,e
B) be a quantum seed in Fand k∈Jex be an
exchangeable index. Define matrices E= (eij )i,j∈Jand F= (fij)i,j∈Jex as follows,
eij =
δij,if j=k,
−1,if i=j=k,
[−bik]+,if i=j=k,
fij =
δij,if i=k,
−1,if i=j=k,
[bkj ]+,if i=k=j.
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 5
Here we write [n]+= max(n, 0) for any n∈Z. Set
µk(e
B) = Ee
BF and µk(Λ) = ETΛE.
Then the pair µk(Λ,e
B) = (µk(Λ), µk(e
B)) is again compatible with the same skew-
symmetrizer D([5, Proposition 3.4]). It is called the mutation of (Λ,e
B)in the
direction k.
Let bk=Pi∈Jbikei∈ZJbe the k-th column vector of e
B. Set
µk(xi) = x−ek+[bk]++x−ek+[−bk]+,if i=k,
xi,if i=k.
Here we write [n]+= ([ni]+)i∈Jfor any n= (ni)i∈J∈ZJ. Then one can show that
the tuple µk(x)=(µk(xi))i∈Jis µk(Λ)-commutative and algebraically independent.
Hence µk(S) = (µk(x), µk(Λ), µk(e
B)) is also a quantum seed in F, called the
mutation of Sin the direction k. Two quantum seeds S′and S′′ are called
mutation-equivalent, denoted by S′∼S′′, if one can be obtained from the other
by a sequence of mutations.
Given a quantum seed Sin F, The quantum cluster algebra Aq(S) is the
Z[q±1/2]-subalgebra of Fgenerated by the union of clusters of all quantum seeds
which are mutation-equivalent to S. The seed Sis called the initial seed of
Aq(S).
Take any quantum seed S′∼S. Let x′= (x′
i)i∈Jbe the cluster of S′. The
quantum Laurent phenomenon ([5, Corollary 5.2]) asserts that the cluster algebra
Aq(S) is contained in the Z[q±1/2]-subalgebra of Fgenerated by (x′
i)±1(i∈Jex)
and x′
i(i∈Jfz). Elements of the form (x′)a(a∈NJ) are called cluster monomials
in Aq(S).
2.2. Quantum groups. In this subsection we recall basic construction in quan-
tum groups following [19] and [15, §1].
We fix once for all an index set Iand a Kac-Moody root datum (A, P, Π, P ∨,Π∨)
which consists of
(a) a symmetrizable generalized Cartan matrix A= (aij)i,j ∈I, that is, an integer
valued matrix such that: (i) aii = 2, for all i∈I, (ii) aij <0, for all i, j ∈I, (iii)
there exists a diagonal matrix D= diag(ti|i∈I) with positive integers tisuch
that DA is symmetric. We normalize (ti)i∈Isuch that they are relatively prime.
(b) a finitely generated free abelian group P, called the weight lattice.
(c) a set of linearly independent elements Π = {αi|i∈I} ⊂ P, called the set
of simple roots.
(d) the dual group P∨= HomZ(P, Z), called the coweight lattice. Let ⟨,⟩:
P∨×P!Zbe the canonical pairing.
(e) a set of elements Π∨={hi|i∈I} ⊂ P∨, called the set of simple coroots,
such that ⟨hi, αj⟩=aij, for i, j ∈I.
We further assume the datum to be simply-connected, that is, for each i∈I,
there exits ϖi∈Psuch that ⟨hj, ϖi⟩=δij for all j∈I. The element ϖiis called
the fundamental weight associated to i.
Set P+={µ∈P| ⟨hi, µ⟩ ≥ 0,∀i∈I}to be the set of dominant weights. The
free abelian group Q=Li∈IZαiis called the root lattice. Set Q+=Li∈IZ≥0αi,
and Q−=Li∈IZ≤0αi. There is a symmetric bilinear form (·,·) on Q⊗ZPsuch
6 JINFENG SONG
that
(αi, αj)=tiaij and ⟨hi, λ⟩=2(αi, λ)
(αi, αi)for λ∈Pand i, j ∈I.
Let gbe the symmetrizable Kac-Moody algebra over Qassociated to the root
datum, and Wbe the Weyl group of gwith generators si(i∈I). Let l(·) be the
length function of W. A reduced expression of wis a sequence (i1, . . . , ir)∈Ir
such that w=si1. . . sirin Wand r=l(w).
The quantum group Uq(g) associated to the root datum is the unital Q(q)-
algebra with generators ei(i∈I), fi(i∈I), and qh(h∈P∨), which subject to
the following relations
q0= 1, qhqh′=qh+h′, qheiq−h=q⟨h,αi⟩ei, qhfiq−h=q−⟨h,αi⟩fi,
eifj−fjei=δij
qtihi−q−tihi
qi−q−1
i
,
1−aij
X
s=0
(−1)s1−aij
si
e1−aij −s
iejes
i= 0,
1−aij
X
s=0
(−1)s1−aij
si
f1−aij −s
ifjfs
i= 0,
for h, h′∈P∨and i, j ∈Iwith i=j.
Here we set
qi=qti,[n]i=qn
i−q−n
i
qi−q−1
i
,[n]i! = [1]i. . . [n]iand n
si
=[n]i!
[n−s]i![s]i!.
Let Uq(n) (resp., Uq(n−)) be the subalgebra of Uq(g) generated by ei’s (resp.,
fi’s). For any i∈I, and n∈N, define divided powers,
e(n)
i=en
i
[n]i!and f(n)
i=fn
i
[n]i!.
The integral form Uq(n) (resp., Uq(n−)) is the Z[q±1]-subalgebra of Uq(n) (resp.,
Uq(n−)) generated by e(n)
i(resp., f(n)
i) for all i∈Iand n∈N.
There is a Q(q)-algebra anti-automorphism ϕon Uq(g) such that
ϕ(ei) = fi, ϕ(fi) = ei,and ϕ(qh) = qh,
for i∈Iand h∈P∨. It is clear that ϕUq(n)=Uq(n−).
Note that Uq(n) has a natural Q+-grading by setting eito be of degree αi(i∈I).
The degree of a homogeneous element xin Uq(n) will be called the weight of x,
denoted by wt(x). For γ∈Q+, we write Uq(n)γto be the subspace of Uq(n)
consisting of homogeneous elements of weight γ.
Let Bbe the canonical basis of Uq(n) ([19, 14.4]). For λ∈Q+, set Bλ⊂Bto
be the subset consisting of elements of weight λ.
Following [19, 1.2.10], we endow the tensor product Uq(n)⊗Q(q)Uq(n) with the
(twisted) algebra structure given by
(x1⊗x2)·(y1⊗y2) = q−(wt(x2),wt(y1))(x1y1⊗x2y2),
for homogeneous elements x1, x2, y1, y2in Uq(n). One can identify the tensor prod-
uct Uq(n)⊗Z[q±1]Uq(n) as a Z[q±1]-subspace of Uq(n)⊗Q(q)Uq(n), which is moreover
aZ[q±1]-subalgebra under the twisted algebra structure.
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 7
Let r:Uq(n)!Uq(n)⊗Q(q)Uq(n) be the algebra homomorphism such that
r(ei) = ei⊗1+1⊗ei,for i∈I.
The map ris called the coproduct on Uq(n). One can show ([19, 1.4.2]) that
r(e(n)
i) =
n
X
s=0
q−s(n−s)
ie(s)
i⊗e(n−s)
i,(4)
for any i∈Iand n∈N. Then we have
rUq(n)⊂Uq(n)⊗Z[q±1]Uq(n).(5)
Define the graded dual of Uq(n) as following,
Aq(n) = M
γ∈Q+
HomQ(q)(Uq(n)γ,Q(q)).
We endow Aq(n) with the Q(q)-algebra structure given by
(φ·ψ)(x)=(φ⊗ψ)(r(x)),
for φ, ψ ∈Aq(n) and x∈Uq(n). This algebra structure is well-defined by weight
consideration.
Define
Aq(n) = {f∈Aq(n)|fUq(n)⊂Z[q±1]}
to be the dual integral form. Thanks to (5), we deduce that Aq(n) is a Z[q±1]-
subalgebra of Aq(n).
By [19, 1.2.3 & 1.2.5] there is a unique symmetric nondegenerate bilinear form
(·,·) : Uq(n)×Uq(n)!Q(q) such that
(a) (1,1) = 1, (ei, ej) = δij (1 −q2
i)−1, for all i, j ∈I;
(b) (x, yy′) = (r(x), y ⊗y′) and (xx′, y) = (x⊗x′, r(y)), for all x, x′, y, y′∈
Uq(n), where the bilinear form on Uq(n)⊗Uq(n) is defined by (x1⊗x2, y1⊗y2) =
(x1, y1)(x2, y2).
Then one has the algebra isomorphism
ι:Uq(n)∼
−! Aq(n),
given by ι(x)(y) = (x, y) for x, y ∈Uq(n).
2.3. Quantized coordinate rings of unipotent subgroups. We denote by Ti
(i∈I) the braid group action T′
i,−1on Uq(g) as in [19, 37.1.3].
Let w∈W. Fix a reduced expression w= (ir, . . . , i1) of w. For any 1 ≤k≤r,
define elements
βk=si1. . . sik−1(αik)∈Q+and eβk=Ti1· · · Tik−1(eik)∈Uq(n)βk.
For any n= (n1, . . . , nr)∈Nr, define elements
ew(n) = e(n1)
β1. . . e(nr)
βr∈Uq(n),
where e(nk)
βk=enk
βk/[nk]ik! (1 ≤k≤r). Also define elements
e′
w(n) = ew(n)
(ew(n), ew(n)) ∈Uq(n) and e∗
w(n) = ι(e′
w(n)) ∈Aq(n).
Let Uq(n(w)) be the Q(q)-subspace of Uq(n) spanned by elements ew(n) (n∈
Nr). It follows from [19, Proposition 40.2.1] and [17, Proposition 4.11] that
8 JINFENG SONG
Uq(n(w)) is a Q(q)-subalgebra of Uq(n) which is independent of the choice of the
reduced expressions of w. Set Uq(n(w)) = Uq(n(w)) ∩Uq(n) to be the integral
form. Then {ew(n)|n∈Nr}is a Z[q±1]-basis of Uq(n(w)), called a PBW basis
associated to the reduced expression w.
Define Aq(n(w)) = ιUq(n(w)). Let Aq(n(w)) = Aq(n)∩Aq(n(w)) be the
integral form. The set {e∗
w(n)|n∈Nr}gives a Z[q±1]-basis of Aq(n(w)), called a
dual PBW basis associated to w.
By [16, Theorem 2.19], one has
Uq(n(w)) = Uq(n)∩Tw−1Uq(n−).
Here Tw−1=Ti1. . . Tir. Also define
Uq(n(w)′) = Uq(n)∩Tw−1Uq(n) and Aq(n(w)′) = ιUq(n(w)′).
Set
Aq(n(w)′) = Aq(n(w)′)∩Aq(n)
to be its integral form. Then Aq(n(w)′) is a free Z[q±1]-module ([16, Theorem
3.3]). Moreover, by [16, Theorem 1.1 (2)], multiplication gives the isomorphism
as Z[q±1]-modules:
Aq(n(w)) ⊗Z[q±1]Aq(n(w)′)∼
−! Aq(n).(6)
2.4. Unipotent quantum minors. For any λ∈P+, let V(λ) be the integrable
simple Uq(g)-module with highest weight λ([15, 1.2]). We fix a nonzero vector
vλ∈V(λ) of weight λ, and set V(λ) = Uq(n−)vλto be the integral form of V(λ).
Then V(λ) is a free Z[q±1]-submodule of V(λ).
For λ∈P+, define I(λ) to be the right ideal of Uq(n) given by
I(λ) = X
i∈I
e(⟨hi,λ⟩+1)
iUq(n).(7)
Set
I′(λ) = ϕI(λ)=X
i∈I
Uq(n−)f(⟨hi,λ⟩+1)
i
to be the left ideal of Uq(n−). Then we have a Z[q±1]-module isomorphism
Uq(n−)/I′(λ)∼
−! V(λ), x +I′(λ)7! xvλ.
For w∈Wwith a fixed reduced expression w= (i1, . . . , ir), define the extremal
vector
vwλ =f(a1)
i1. . . f (ar)
irvλ∈V(λ),
where at=⟨hit, sit+1 . . . sir(λ)⟩, for 1 ≤t≤r. By the quantum Verma identity
([19, Proposition 39.3.7]), the element vwλ only depends on the weight wλ, and
not on the choice of wand its reduced expressions.
Following [19, Proposition 19.1.2], there is a unique non-degenerate symmetric
bilinear form (·,·)λ:V(λ)×V(λ)!Q(q) such that
(vλ, vλ)λ= 1 and (xu, v)λ= (u, ϕ(x)v)λ,
for x∈Uq(g) and u, v ∈V(λ). Moreover, the bilinear form restricts to the integral
form: (·,·)λ:V(λ)×V(λ)!Z[q±1] ([19, 19.3.3 (c)]).
By definition, for any λ∈P+,w∈Wand x∈I(λ), we have
(xvwλ, vλ)λ= (vwλ, ϕ(x)vλ)λ= 0.(8)
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 9
For λ∈P+and w, u ∈W, define the unipotent quantum minor D(wλ, uλ) in
Aq(n) by
D(wλ, uλ)(x) = (xvwλ , vuλ)λ,for x∈Uq(n).
It is proved in [9, Proposition 6.3] that any (nonzero) unipotent quantum minor
is a dual canonical basis element which can be described explicitly using the theory
of crystal basis. The following lemma is a special case of loc. cit.
Lemma 4. For any λ∈P+and w∈W, there is a unique element bin Bλ−wλ
such that b∈ I(λ). One has
b∈e(a1)
i1. . . e(ar)
ir+I(λ),
where w= (ir, . . . , i1)is any reduced expression of w, and at=⟨hit, sit−1. . . si1(λ)⟩,
for 1≤t≤r. Moreover we have D(wλ, λ)(b′) = δb,b′for any b′∈B.
2.5. Quantum cluster algebra structures. We describe the quantum cluster
algebra structure on the quantized coordinate rings in this subsection. To do that
we firstly need to extend coefficients to Z[q±1/2]. Here q1/2is a square root of qin
the algebraic closure of Q(q). Set
Aq1/2(n) = Z[q±1/2]⊗Z[q±1]Aq(n) and Uq1/2(n) = Z[q±1/2]⊗Z[q±1]Uq(n).
We always view Aq(n) (resp., Uq(n)) as a subset of Aq1/2(n) (resp., Uq1/2(n)).
Take any w∈W. Set
Aq1/2(n(w)) = Z[q±1/2]⊗Z[q±1]Aq(n(w)) ⊂Aq1/2(n).
Since Aq1/2(n(w)) is an Ore domain, one can embed Aq1/2(n(w)) into its skew field
of fractions, which we denote by Fq1/2(n(w)) ([9, §7.5]).
Fix a reduced expression w= (ir, . . . , i1) of w. For 1 ≤t≤r, define
λt=si1. . . sit(ϖit) and Dt=D(λt, ϖit).(9)
Then Dt(1 ≤t≤r) belongs to Aq1/2(n(w)) ([9, Corollary 12.4] [12, Theorem
7.3]).
Let J={1, . . . , r}. Set
Jfz ={k∈J|is=ik,∀s>k}and Jex =J\Jfz.
Note that for k∈Jfz one has λk=w−1(ϖik).
By [5, Theorem 10.1], we can take an integer-valued skew-symmetric matrix
Λ=(λtk)t,k∈Jsuch that
DtDk=qλtk DkDt.
For t∈J, set
t+= min({k|t < k ≤r, ik=it}∪{r+ 1}),
t−= max({k|1≤k < t, it=ik}∪{0}).
Following [12, Proposition 7.2], define the integer-valued J×Jex-matrix e
Bw,
(e
Bw)tk =
1,if t=k−,
−1,if t=k+,
aitik,if t<k<t+< k+,
−aitik,if k<t<k+< t+,
0,otherwise.
10 JINFENG SONG
By loc.cot., we have:
(a) The pair (Λ,e
Bw)is a compatible pair with the skew-symmetrizer D, where
the Jex ×Jex part of Dis the diagonal matrix with diagonal entries dk= 2tik
(k∈Jex).
Theorem 5. ([12, 15]) There exists some γt∈1
2Z, for any t∈J, such that
the triple Sw= (x= (qγtDt)t∈J,Λ,e
Bw)is a quantum seed in Fq1/2(n(w)), and
moreover one has
Aq(Sw) = Aq1/2(n(w)).
2.6. Change of rings. Let Rbe any (unital) commutative Z[q±1/2]-algebra. Set
UR(n) = R⊗Z[q±1/2]Uq1/2(n) and AR(n) = R⊗Z[q±1/2]Aq1/2(n).
It is clear that UR(n) has a Q+-grading and AR(n) can be identified with the
graded dual, that is,
AR(n)∼
=M
γ∈Q+
HomR(UR(n)γ, R).
Take w∈Wand define
AR(n(w)) = R⊗Z[q±1/2]Aq1/2(n(w)).
Since Aq(n(w)) is spanned by a subset of dual canonical basis ([17, Theorem
4.25]), the quotient Aq1/2(n)/Aq1/2(n(w)) is a free Z[q±1/2]-module. Hence we can
identify AR(n(w)) as an R-subalgebra of AR(n).
2.7. Quantum Frobenius morphisms and their splittings. We recall the
construction of quantum Frobenius morphisms and their splittings in this subsec-
tion.
Let lbe a positive odd integer which is coprime to ti, for i∈I. Let εbe a
primitive l-th root of unity. Define Z[q±1/2]-algebra R1,Rεas following. We set
R1=Rε=Z[ε] as rings, and the Z[q±1/2]-algebra structure on R1(resp., Rε) is
given by q1/27! 1 (resp., q1/27! ε(l+1)/2).
Recall §2.6. We write U1(n) = UR1(n), Uε(n) = URε(n), A1(n) = AR1(n) and
Aε(n) = ARε(n) for simplicity.
For any element x∈Aq1/2(n), write
εx= 1 ⊗Z[q±1/2]x∈Aε(n) and 1x= 1 ⊗Z[q±1/2]x∈A1(n).
Similarly, for any y∈Uq1/2(n), write
εy= 1 ⊗Z[q±1/2]y∈Uε(n) and 1y= 1 ⊗Z[q±1/2]y∈U1(n).
The first part of the following theorem is originally due to Lusztig [18] [19,
35.1.7] with certain additional assumptions on l, and McGerty [23] gave another
construction using Hall algebras where the additional assumptions were removed.
The second part is due to Lusztig [19, 35.1.8].
Theorem 6. (a) There exists a unique Z[ε]-algebra homomorphism
Fr :Uε(n)−! U1(n)
such that for i∈Iand n∈N,Fr(εe(n)
i)equals 1e(n/l)
iif ldivides n, and equals 0
otherwise.
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 11
(b) There exists a unique Z[ε]-algebra homomorphism
Fr′:U1(n)−! Uε(n)
such that Fr′(1e(n)
i) = εe(nl)
i, for i∈Iand n∈N.
Now we pass to the dual side. Define
Fr∗:A1(n)−! Aε(n) and Fr′,∗:Aε(n)−! A1(n),
by
Fr∗(f)(x) = f(Fr(x)) for f∈A1(n), x ∈Uε(n),
and
Fr′,∗(g)(y) = g(Fr′(y)) for g∈Aε(n), y ∈U1(n).
These two maps are well-defined Z[ε]-linear maps by weight consideration.
It follows immediately from definitions that these two maps are compatible with
Q+-grading, that is,
Fr∗A1(n)µ⊂Aε(n)lµ and Fr′,∗Aε(n)µ⊂A1(n)µ/l ,(10)
for any µ∈Q+. Here A1(n)µ/l is understood as zero space if µ∈ lQ+.
The following proposition was obtained in [8, Proposition 4] when gis semisim-
ple.
Proposition 7. The map Fr∗is a Z[ε]-algebra homomorphism. The map Fr′,∗
satisfies
Fr′,∗◦Fr∗=id and Fr′,∗Fr∗(f)g=fFr′,∗(g),
for f∈A1(n)and g∈Aε(n).
Proof. Let rε:Uε(n)!Uε(n)⊗Uε(n) and r1:U1(n)!U1(n)⊗U1(n) be the
base changes of the coproduct r, where the tensor products are over Z[ε]. The
twisted algebra structure naturally extends to Uε(n)⊗Uε(n) and U1(n)⊗U1(n).
Then rεand r1are both algebra homomorphisms.
By definition, the map Fr∗is an algebra homomorphism if and only if we have
(Fr ⊗Fr)◦rε=r1◦Fr (11)
as maps from Uε(n) to U1(n)⊗U1(n). Thanks to (4) it is direct to check (11) when
acting on divided powers. Since maps Fr,r1and rεare all algebra homomorphisms,
it will suffice to show Fr ⊗Fr is also an algebra homomorphism (with respect to
the twisted product structures on tensor products).
Take x1,x2,y1and y2to be any homogeneous elements in Uε(n). Then
(Fr ⊗Fr)(x1⊗x2)·(y1⊗y2)=ε−(wt(x2),wt(y1))Fr(x1y1)⊗Fr(x2y2)
(♡)
=Fr(x1y1)⊗Fr(x2y2)
= (Fr ⊗Fr)(x1⊗x2)·(Fr ⊗Fr)(y1⊗y2).
The equality (♡) follows by noticing that both sides are zero unless wt(x2) and
wt(y1) both belong to lQ+. Therefore equality (11) holds and Fr∗is an algebra
homomorphism.
12 JINFENG SONG
We prove the second statement. Since Fr◦Fr′=id, we have Fr′,∗◦Fr∗=id. By
definition, the last equality holds if and only if the following diagram commutes
U1(n)Uε(n)Uε(n)⊗Uε(n)
U1(n)⊗U1(n)U1(n)⊗Uε(n).
Fr′
r1
rε
Fr⊗id
id⊗Fr′
(12)
Here tensor products are over Z[ε]. We endow U1(n)⊗Uε(n) with the natural
algebra structure (without twisting), and claim that all the maps in the diagram
(12) are algebra homomorphisms. We only prove this claim for the map Fr ⊗id.
The proof for other maps is trivial.
Take homogeneous elements xi,yi(i= 1,2) in Uε(n). Then
(Fr ⊗id)(x1⊗x2)·(y1⊗y2)= (Fr ⊗id)(ε−(wt(x2),wt(y1))x1y1⊗x2y2)
=Fr(x1y1)⊗(x2y2)
= (Fr ⊗id)(x1⊗x2)·(Fr ⊗id)(y1⊗y2),
where the second equality follows by noticing that both sides are zero unless
wt(y1)∈lQ+.
It then remains to check the diagram (12) when acting on divided powers, which
is direct and will be skipped. □
It follows from the above proposition that we can endow Aε(n) with the A1(n)-
module structure via the map Fr∗. Then Fr′,∗is a splitting of the map Fr∗as
A1(n)-module homomorphisms.
2.8. Proof of the main theorem. We prove Theorem 1 in this subsection.
Retain the same notation as in §2.6 and §2.7.
Lemma 8. Suppose Ris an integral domain and wis a Weyl group element.
Then we have:
(a) the R-algebra AR(n(w)) is an integral domain;
(b) for any f∈AR(n), if g·f∈AR(n(w)) for some nonzero element gin
AR(n(w)), then f∈AR(n(w));
(c) for any f∈AR(n), if g·f= 0 for some nonzero element gin AR(n(w)),
then f= 0.
Proof. Since AR(n(w)) is a free R-module, it will suffice to prove the lemma over
its field of fractions. We may assume Ris a field.
Take a reduced expression w= (ir, . . . , i1) and use the same notations as in
§2.3. For n,m∈Zr, it follows from the dual Levendorskii–Soibelman formula
[17, Theorem 4.27] that
e∗
w(n)e∗
w(m)∈qZe∗
w(n+m) + X
n′<n+m
Z[q±1]e∗
w(n′) in Aq(n(w)). (13)
Here <is the lexicographic order on Zr.
Take fand gto be two nonzero elements in AR(n(w)). By rescaling, we may
assume that
f∈1⊗e∗
w(n1) + X
n′<n1
R(1 ⊗e∗
w(n′)), g ∈1⊗e∗
w(n2) + X
n′<n2
R(1 ⊗e∗
w(n′)).
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 13
Then by (13), we have
fg ∈1⊗e∗
w(n1+n2) + X
n′<n1+n2
R(1 ⊗e∗
w(n′)).
In particular, we deduce that fg is nonzero, so AR(n(w)) is an integral domain.
This proved the assertion (a).
The assertion (b) and (c) follow from (a) and the tensor product decomposition
(6). □
Lemma 9. For any f, g ∈Aε(n), we have Fr′,∗(f g) = Fr′,∗(gf ).
Proof. Let rε:Uε(n)!Uε(n)⊗Uε(n) be the base change of the coproduct r
as before, and s:Uε(n)⊗Uε(n)!Uε(n)⊗Uε(n) be the linear map given by
s(x⊗y) = y⊗x. Here the tensor products are all over Z[ε].
By definition it will suffice to prove that
rε◦Fr′(x) = s◦rε◦Fr′(x),(14)
for any x∈U1(n).
Let Mbe the Z[ε]-subspace of Uε(n)⊗Uε(n) spanned by elements of the form
x⊗y, where xand yare homogeneous elements in Uε(n) such that the summation
wt(x) + wt(y) belongs to lQ+. It is clear that the subspace Mis moreover a
subalgebra and s(M) = M.
Notice that for any homogeneous elements x1,x2,y1and y2in Uε(n) with
wt(x1) + wt(x2)∈lQ+and wt(y1) + wt(y2)∈lQ+, we have
s((x1⊗x2)·(y1⊗y2)) = ε−(wt(x2),wt(y1))x2y2⊗x1y1
=ε−(wt(x1),wt(y2))(x2y2⊗x1y1) = s(x1⊗x2)·s(y1⊗y2).
Therefore s|M:M!Mis an algebra homomorphism.
It is direct to see that rε◦Fr′(U1(n)) ⊂M. Hence both s◦rε◦Fr′and rε◦Fr′are
algebra homomorphisms. It will suffice to check (14) for xbeing divided powers,
which follows immediately from (4). □
For any w∈W, set Aε(n(w)) = ARε(n(w)) and A1(n(w)) = AR1(n(w)). Fix a
reduced expression w= (ir, . . . , i1) of w. Let J={1, . . . , r}. Recall the partition
J=Jex ⊔Jfz in §2.5 and the unipotent quantum minors Dt∈Aq(n(w)) (t∈J) in
§2.4.
Recall the initial quantum seed Sw= (x= (qγtDt)t∈J,Λ = (λtk)t,k∈J,e
Bw) of
Aq1/2(n(w)) in §2.5. In particular, the tuple D= (Dt)t∈Jis Λ-commutative. For
a= (at)t∈J∈ZJ, define the element
Da=q1/2Pi>j aiajλij Da1
1Da2
2· · · Dar
rin Fq1/2(n(w)).
For any a∈ZJ, write a/l ∈ZJto be the tuple which equals 0if a∈ lZJand
equals (at/l)t∈Jif a∈lZJ.
Proposition 10. For any a∈NJ, we have
Fr∗1Da=εDlaand Fr′,∗εDa=1Da/l.
Remark 11. When gis semisimple, the above formula of Fr∗was obtained in
[8, Proposition 4]. When gis semisimple and abelongs to lNJ, the above formula
14 JINFENG SONG
of Fr′,∗was also obtained in loc. cit.. We proved the formulas when gis Kac-
Moody. The vanishing of the element Fr′,∗εDafor anot belonging to lNJis new
even when gis semisimple.
Proof of Proposition 10. The case when w=eis trivial. We assume w=e.
For any λ∈P+, write
Vε(λ) = Rε⊗Z[q±1]V(λ) and V1(λ) = R1⊗Z[q±1]V(λ).
Let ε·,·λ(resp., 1·,·λ) be the Z[ε]-bilinear form on Vε(λ) (resp., V1(λ)) ob-
tained from base change of ·,·λ:V(λ)×V(λ)!Z[q±1].
We then prove the first equality. Notice that by (2), we have
Dl
tDl
t′=ql2Λ(et,et′)Dl
t′Dl
tin Aq1/2(n),
for t, t′∈J. Here ek(k∈J) is the standard basis element in ZJ. Therefore after
specialization, element εDl
t(t∈J) commutes with each other in Aε(n). Since Fr∗
is an algebra homomorphism, it will suffice to show for any t∈J,
Fr∗(1Dt) = εDl
t.(15)
By [15, Corollary 9.1.3], one has
Dl
t=D(λt, ϖit)l=q−l(l−1)/2·(ϖit,ϖit−λt)D(lλt, lϖit) in Aq1/2(n).
Here λt=si1. . . sitϖit=w−1ϖitas in (9).
Hence in Aε(n),
εDl
t=εD(lλt, lϖit).
By Lemma 4, take b∈Bsuch that D(lλt, lϖit)(b′) = δb,b′, for b′∈B. Then
after base change we have
εDl
t(εb′) =εD(lλt, lϖit)(εb′) = δb,b′,
for b′∈B.
One the other hand, the element Fr∗(1Dt)∈Aε(n) is given by
Fr∗(1Dt)(f) = (1Dt)(Fr(f)) = 1Fr(f)vλt, vϖitϖit
,for f∈Uε(n).
Here we use the same notations to denote images of extremal vectors after base
changes. For any b′∈B, it follows from weight considerations that Fr∗(1Dt)(1b′) =
0 unless wt(b′) = lϖit−lλt.
Recall the left ideal I(λ) (λ∈P+) from (7). By definition, one has
FrRε⊗I(lϖit)⊂R1⊗I(ϖit).(16)
Take b′∈Blϖit−lλt. If b′=b, by Lemma 4 we have b′∈I(lϖit). Combining (16)
and (8), we have
Fr∗(1Dt)(εb′) = 1Fr(εb′)vλt, vϖitϖit
= 0.
By Lemma 4, we have
b∈e(lat)
it. . . e(la1)
i1+I(lϖit),
where ak=⟨hik, sik−1. . . sit(ϖit)⟩, for 1 ≤k≤t. Hence by (16) and (8), we have
Fr∗(1Dt)(εb) = 1Fr(εb)vλt, vϖitϖit
=11(e(at)
it. . . e(a1)
i1)vλt, vϖitϖit
= 1.
In conclusion we have Fr∗(1Dt)(εb′) = εDl
t(εb′) for b′∈B. Therefore (15) is proved.
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 15
We next prove the second equality. Firstly suppose a=la′for some a′∈Nr.
Then thanks to Proposition 7 and the previous discussion, we have
Fr′,∗(εDa) = Fr′,∗◦Fr∗(1Da′) = 1Da′.
Now suppose a∈ lNr. We need to show that Fr′,∗(εDa) = 0.
Recall Jfz ={t∈[1, r]|ik=it,∀k > t}and Jex = [1, r]\Jfz. We divide into two
cases depending on divisibility of at(t∈Jex) by l.
Case I: l|at, for all t∈Jex .
Write a=la′+a′′, with a′= (a′
t)t∈J,a′′ = (a′′
t)t∈Jin NJand 0 ≤a′′
t< l
(1 ≤t≤r). Then a′′
t= 0 for t∈Jex and a′′ =0by our assumption. By (2) we
have
Dla′Da′′ =q1/2Λ(la′,a′′)Dain Aq1/2(n).
Hence after base change we have εDa=εDla′
εDa′′ in Aε(n).
It then follows from Proposition 7 that
Fr′,∗(εDa) = Fr′,∗(εDla′
εDa′′ ) = Fr′,∗(Fr∗(1Da′)εDa′′) = 1Da′·Fr′,∗(εDa′′).
Hence it will suffice to show Fr′,∗(εDa′′ ) = 0.
Recall λt=si1. . . sitϖit=w−1ϖitfor t∈Jfz. Since a′′ only contains the et
factors for those t∈Jfz, the monomial Da′′ is a product of the quantum minors of
the form D(a′′
t(w−1ϖit), a′′
tϖit) for various t∈Jfz. It follows from [15, Corollary
9.1.3] that
εDa′′ =εNεD(w−1µ, µ) in Aε(n)
for some N∈Z, and µ∈P+where ⟨hi, µ⟩< l, for i∈I.
Take any f=e(b1)
j1. . . e(bm)
jmin Uq(n). We have
Fr′,∗(εD(w−1µ, µ))(1f) = 1Fr′(f)vw−1µ, vµµ
=1e(lb1)
j1. . . e(lbm)
jmvw−1µ, vµµ
=1vw−1µ, f (lbm)
jm. . . f (lb1)
j1vµµ.
By our condition on µ, the vector f(lbm)
jm. . . f (lb1)
j1vµequals 0 unless b1=· · · =bm=
0, in which case f= 1. In any cases, the element e(lb1)
j1. . . e(lbm)
jmvw−1µ, vµµequals
0. Therefore we deduce that Fr′,∗(εDa′′) = 0.
Case II:l∤at, for some t∈Jex. We fix such an index t.
Write e
Bw= (bij)i∈J,j∈Jex . Let bt=Pr
i=1 bitei∈ZJbe the t-th column vector.
Since (Λ,e
Bw) is compatible with skew-symmetrizer described in §2.5 (a), we have
Λ(bt,ek) = 2tikδtk for 1 ≤k≤r.
Take b′
t∈NJsuch that b′
t−bt∈lZJ. Take a′∈NJsuch that c=a+la′−b′
t∈NJ.
Then we have
DcDb′
t−Db′
tDc= (qΛ(c,b′
t)−q−Λ(c,b′
t))Da+la′in Aq1/2(n).
Hence by Lemma 9 and Proposition 7 we have in Aε(n)
0 = Fr′,∗(εDcεDb′
t−εDb′
tεDc)=(εΛ(c,b′
t)−ε−Λ(c,b′
t))Fr′,∗(εDa+la′)
= (εΛ(c,b′
t)−ε−Λ(c,b′
t))(1Da′)Fr′,∗(εDa).(17)
16 JINFENG SONG
Notice that modulo lwe have
Λ(c,b′
t) = Λ(a−b′
t,b′
t) = Λ(a,b′
t) = Λ(a,bt) = −2titat,
which is nonzero since (l, 2tit) = 1 and l∤atby our assumption.
Therefore εΛ(c,b′
t)−ε−Λ(c,b′
t)= 0. Since 1Da′is a nonzero element in A1(n(w)),
by Lemma 8 (c) and (17) we deduce that Fr′,∗(εDa) = 0. We finish the proof. □
Corollary 12. For any w∈W, we have
Fr′,∗Aε(n(w))⊂A1(n(w)) and Fr∗A1(n(w))⊂Aε(n(w)).
Proof. Retain the same notations as before. Take any f∈Aq1/2(n(w)). By the
Laurent phenomenon, we have
f∈Z[q±1/2]⟨D±1
t|1≤t≤r⟩in Fq1/2(n(w)).
We take a∈Nrwith entries large enough such that
Dlaf∈Z[q±1/2]⟨Dt|1≤t≤r⟩in Aq1/2(n(w)).
Then in Aε(n(w)), the product εDlaεfis a Z[ε]-linear combination of elements in
the form εDa(a∈Nr). By Proposition 7 and Proposition 10, we deduce that
(1Da)Fr′,∗(εf) = Fr′,∗(Fr∗(1Da)εf) = Fr′,∗(εDlaεf)∈A1(n(w)).
Note that 1Dabelongs to A1(n(w)). By Lemma 8 (b) we deduce that Fr′,∗(εf)
also belongs to A1(n(w)). Hence we have Fr′,∗Aε(n(w))⊂A1(n(w)).
One can show another inclusion in a similar way by using the fact that Fr∗is
an algebra homomorphism. □
We are now ready to prove the main theorem.
Proof of Theorem 1. Let S=µks. . . µk1(Sw) be a quantum seed of Aq1/2(n(w))
which is mutation-equivalent to the initial seed Sw, where (ks, . . . , k1) is a sequence
of exchangeable indices. We prove the theorem by induction on s.
Base case: Suppose s= 0. Then S=Sw= (x= (xt)t∈J,Λ,e
Bw), where
xt=qγtDtwith γt∈1
2Z. Take a= (at)t∈J∈NJ. Thanks to Proposition 10, we
have
Fr∗(1xa) = Fr∗(1Da) = εDla=εxla
and Fr′,∗(εxa) = Y
t∈J
εatγtFr′,∗(εDa)(♡)
=1Da/l =1xa/l.
Here (♡) follows by noticing that both sides are zero unless a∈lQ+, in which
case εatγt= 1 for t∈J. Hence the theorem is proved for s= 0.
Induction step: Suppose the theorem holds for the quantum seed S′= (x′=
(x′
t)t∈J,Λ′,e
B′). We show it holds for the quantum seed S′′ =µk(S′)=(x′′ =
(x′′
t)t∈J,Λ′′,e
B′′) for any k∈Jex.
Take a= (at)t∈J∈NJ. We firstly prove Fr∗(1(x′′)a) = ε(x′′ )la,for a∈NJ.
Since Fr∗is an algebra homomorphism, it suffices to show that
Fr∗(1x′′
t) = ε(x′′
t)lfor t∈J. (18)
Suppose t=k. Then x′′
t=x′
t, and (18) follows immediately from our assump-
tion.
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 17
Now suppose t=k. Let b′
k=Pr
i=1 b′
ikei∈ZJbe the k-th column vector. Here
b′
ik is the (i, k)-entry of e
B′as usual. By definition, we have
x′′
k= (x′)−ek+[b′
k]++ (x′)−ek+[−b′
k]+in Fq(n(w)).
By §2.5 (a) we have
Λ′(−ek+ [b′
k]+,−ek+ [−b′
k]+) = Λ′(b′
k,−ek+ [−b′
k]+) = −2tik.
One has
(x′)−ek+[b′
k]+(x′)−ek+[−b′
k]+=q−2tik(x′)−ek+[−b′
k]+(x′)−ek+[b′
k]+.
By the quantum binomial identity ([19, 1.3.5]), in Fq1/2(n(w)) we have
(x′′
k)l=
l
X
t=0
q−t(l−t)tikl
tq−tik
(x′)−tek+t[b′
k]+(x′)−(l−t)ek+(l−t)[−b′
k]+
=
l
X
t=0
q−t(l−t)tik+1/2Λ′(−tek+t[b′
k]+,−lek+t[b′
k]++(l−t)[−b′
k]+)l
tq−tik
(x′)−lek+t[b′
k]++(l−t)[−b′
k]+.
Multiplying (x′
k)lon both sides, we get the following equation in Aq1/2(n),
(x′
k)l(x′′
k)l
=
l
X
t=0
q−t(l−t)tik+1/2Λ′((l−t)ek+t[b′
k]+,−lek+t[b′
k]++(l−t)[−b′
k]+)l
tq−tik
(x′)t[b′
k]++(l−t)[−b′
k]+.
Recall the Z[q±1/2]-module structure on Rεwhere q1/2acts by ε1/2=ε(l+1)/2. Since
2tikis coprime to the odd number l, the number ε−tikis a primitive l-th root of
unity. By [19, 34.1.2], in Rεwe have
l
tq−tik
= 0 unless l|t.
Therefore in Aε(n) we have
ε(x′
k)l
ε(x′′
k)l=ε(x′)l[−b′
k]++ε(x′)l[b′
k]+.(19)
On the other hand, in A1(n) we have
1(x′
k)1(x′′
k) = 1(x′)[−b′
k]++1(x′)[b′
k]+.
Applying Fr∗and using our assumption, we have
ε(x′
k)lFr∗(1x′′
k) = ε(x′)l[−b′
k]++ε(x′)l[b′
k]+.
Combining with (19), we have
ε(x′
k)l(Fr∗(1x′′
k)−εx′′
k) = 0.(20)
Thanks to Corollary 12, the element Fr∗(1x′′
k) belongs to Aε(n(w)). Hence (20)
holds in Aε(n(w)). Since Aε(n(w)) is an integral domain by Lemma 8 (a), we
deduce that Fr∗(1x′′
k) = εx′′
k.Therefore (18) is proved.
We next prove Fr′,∗ε(x′′)a) = 1(x′′ )a/l, for a= (ak)k∈J∈NJ. Take a′= (a′
t)t∈J
such that a=a′+akek. Then a′
k= 0.
Firstly suppose l|ak. In Aq1/2(n) one has
(x′′)a=q−1/2Λ(akek,a)(x′′
k)ak(x′′)a′=q−1/2Λ(akek,a)(x′′
k)ak(x′)a′.
18 JINFENG SONG
We deduce that
Fr′,∗(ε(x′′ )a) = ε−1/2Λ(akek,a)Fr′,∗(ε(x′′
k)akε(x′)a′)
=Fr′,∗(Fr∗(1(x′′
k)ak/l)ε(x′)a′)
=1(x′′
k)ak/lFr′,∗(ε(x′)a′)
=1(x′′
k)ak/l1(x′)a′/l
=1(x′′
k)ak/l1(x′′ )a′/l =1(x′′)a/l .
Next suppose l∤ak. We need to show that Fr′,∗(ε(x′′)a) = 0. Notice that in
Fq1/2(n(w)) the product (x′
k)lc(x′′ )a(c∈Z) is a Z[q±1/2]-linear combination of
monomials (x′)(−ak+lc)ek+rfor various r= (rt)t∈Jwith rk= 0. Choose c > 0 such
that −ak+lc > 0. Then each term (x′)(−ak+lc)ek+rbelongs to Aq1/2(n). Hence
in Aε(n) the product ε(x′
k)lcε(x′′ )ais a Z[ε]-linear combination of elements of the
form ε(x′)(−ak+lc)ek+r, for various ras before. Since −ak+lc is not divided by l
and rdoes not have ekfactor, we deduce that Fr′,∗(ε(x′)(−ak+lc)ek+r) = 0 by our
assumption. Therefore
0 = Fr′,∗(ε(x′
k)lcε(x′′ )a) = Fr′,∗(Fr∗(1(x′
k)c)ε(x′′)a) = 1(x′
k)cFr′,∗(ε(x′′ )a).
Since A1(n) is an integral domain by Lemma 8 (a), we deduce that Fr′,∗(ε(x′′)a) =
0, which completes the proof. □
Remark 13. Analogous computations in the induction step also show up in [13,
26], where the authors used them to construct a canonical central subalgebra and
a regular trace map for a root of unity quantum cluster algebra.
3. Frobenius splittings of unipotent subgroups
Retain the same notations as in the previous section. We further assume that
l=pis a prime number. Fix an algebraically closed field kwith characteristic p.
In this section, a scheme will always mean a separated scheme of finite type
defined over k. A variety is a reduced scheme. We will not distinguish a variety
with its k-rational points.
3.1. Frobenius splittings. Following [4], we recall the concept of Frobenius split-
tings in this subsection.
For a scheme X, the absolute Frobenius morphism
F=FX:X−! X
is the identity map on the underlying space, and the p-th power map on the
structure sheaf OX.
AFrobenius splitting of Xis a morphism (as sheaves of abelian groups)
φ:OX−! OX
such that
(a) φ(fpg) = fφ(g) for any local sections f, g ∈ OX, and
(b) φ(1) = 1.
Note that (a) is equivalent to the requirement that φ∈HomOX(F∗OX,OX).
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 19
Assume Xis nonsingular, and let ωXbe the canonical sheaf of X. One has the
canonical isomorphism ([4, §1.3])
HomOX(F∗OX,OX)∼
=H0(X, ω1−p
X) (21)
as OX(X)-modules.
Let Ybe a closed subscheme of X. A Frobenius splitting φof Xis said to
compatibly split Yif φpreserves the ideal sheaf defining Y.
Let Rbe a (finitely generated) k-algebra and X= Spec R. Denote by EndF(R)
the space of additive maps φ:R!R, such that φ(fpg) = fφ(g) for any f, g ∈R.
Then one has canonical isomorphisms as R-modules,
EndF(R)∼
=HomOX(F∗OX,OX).(22)
A Frobenius splitting of X= Spec Ris equivalent to a map φ∈EndF(R) such
that φ(1) = 1. Such a map will also be called a Frobenius splitting of the k-algebra
R.
3.2. Algebraic Frobenius splittings. We endow kwith the Z[q±1/2]-module
structure via q1/27! 1. Fix w∈W. Let
Ak(n(w)) = k⊗Z[q±1/2]Aq1/2(n(w)).
We write kf= 1 ⊗f∈Ak(n(w)), for any f∈Aq1/2(n(w)).
Recall that εis a primitive p-th root of unity. We further endow kwith the Z[ε]-
algebra structure via ε7! 1. Then we have canonical isomorphisms as k-algebras
Ak(n(w)) ∼
=k⊗Z[ε]Aε(n(w)) ∼
=k⊗Z[ε]A1(n(w)).
Suppse l(w) = r. Fix a reduced expression w= (ir, . . . , i1) of w. Recall from
§2.3 the dual PBW basis {e∗
w(n)|n∈Nr}of Aq1/2(n(w)). Write yt∈Ak(n(w))
to be the image of dual PBW basis element corresponding to n=etfor 1 ≤t≤r.
Here et(1 ≤t≤r) is the standard basis element of Zras usual.
We endow Ak(n(w)) with a Q+-grading given by setting wt(yt) = βtwhere
βt=si1. . . sit−1(αit)∈Q+for 1 ≤t≤r. The degree of a homogeneous element
in Ak(n(w)) will be called its weight as before.
The following claim follows from the dual Levendorskii–Soibelman formula (13)
and the fact that {e∗
w(n)|n∈Nr}is a basis.
(a) Elements yt(1≤t≤r) are algebraically independent in Ak(n(w)). More-
over these elements generate Ak(n(w)) as a k-algebra.
In other words one has
Ak(n(w)) = k[y1, . . . , yt] (23)
as k-algebras. The identity (23) gives an isomorphism as varieties
Spec Ak(n(w)) ∼
=Ar.(24)
This isomorphism depends on the choice of the reduced expressions w.
Recall from Corollary 12 that Fr∗and Fr′,∗restrict to subalgebras, that is,
Fr∗|A1(n(w)) :A1(n(w)) !Aε(n(w)),Fr′,∗|Aε(n(w)):Aε(n(w)) !A1(n(w)).
20 JINFENG SONG
Set
Fw= (·)p⊗Z[ε]Fr∗|A1(n(w)) :Ak(n(w)) −! Ak(n(w)),
φw= (·)1/p ⊗Z[ε]Fr′,∗|Aε(n(w)) :Ak(n(w)) −! Ak(n(w)),
where (·)pand (·)1/p are automorphisms on kgiven by t7! tpand t7! t1/p,
respectively.
Proposition 14. The map Fwis the p-th power map on Ak(n(w)), and the map
φwis a Frobenius splitting of the k-algebra Ak(n(w)).
Moreover, for any quantum cluster monomial xa∈Aq1/2(n(w)) (a∈Nr), we
have
φw(kxa) = kxa/p.
Proof. The assertion that Fwis the p-th power map follows from the similar argu-
ment as in [6, Proposition 5.2 (1)]. The fact that φwis a Frobenius splitting follows
from Proposition 7. The remaining part is a direct consequence of Theorem 1. □
The Frobenius splitting φwwill be called the algebraic splitting of Ak(n(w)).
The next two subsections are devoted to show that the algebraic splittings coincide
with the canonical splittings on Schubert cells in finite types.
3.3. Geometric description of the algebraic splittings. We write N(w) =
Spec Ak(n(w)). Then the map φwgives a Frobenius splitting of N(w). The goal
of this subsection is to describe the algebraic splitting φwunder the isomorphism
(21).
Thanks to (24), the canonical sheaf of N(w) is trivial. Take a nowhere vanishing
global section v∈H0(N(w), ωN(w)).Note that the choice of vis unique up to k∗-
rescaling.
By (22) we have isomorphisms
EndF(Ak(n(w))) ∼
=H0(N(w), ω1−p
N(w)) = Ak(n(w))v1−p(25)
as Ak(n(w))-modules.
Set
Supp(w) = {i∈I|si≤win the Bruhat order}
to be the set of indices of the support of w. For i∈Supp(w), the quantum
unipotent minor D(w−1ϖi, ϖi) is a frozen cluster variable in Aq1/2(n(w)) up to a
power of q. Therefore the element
pi=kD(w−1ϖi, ϖi)∈Ak(n(w))
is a prime element in Ak(n(w)). Set
Yi={x∈N(w)|pi(x) = 0}
to be the prime divisor of N(w) defined by pi. It corresponds to the Richardson
divisor of the Schubert cell.
Proposition 15. The splitting φwof N(w)compatibly splits divisors Yi, for i∈
Supp(w).
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 21
Proof. Take (xt)t∈Jto be a cluster of Aq1/2(n(w)). Let Tbe the k-subalgebra
of the fractional field of Ak(n(w)), generated by kxt(t∈Jfz) and k(x±1
t′) (t′∈
Jex). By the Laurent phenomenon, the algebra Ak(n(w)) is contained in T. Let
U= Spec T. Then Uis an open dense subset of N(w). The Frobenius splitting
φwrestricts a splitting of U. Thanks to Proposition 14, this splitting compatibly
splits subvarieties (of U) defined by kxt|U= 0 for any t∈Jfz . By construction we
have Y
t∈Jfz
kxt=Y
i∈Supp(w)
pi.
Hence φw|Ucompatibly splits Yi∩Ufor any i∈Supp(w). By [4, Lemma 1.1.7
(ii)] we complete the proof. □
Let k[P] be the group algebra (over k) of the weight lattice P, and Hbe the
algebraic torus associated to k[P]. The Q+-grading on Ak(n(w)) gives an algebraic
action of the torus Hon N(w). Then Halso acts on the space EndF(Ak(n(w))
as k-linear maps, given by
(h∗φ)f=h(φ(h−1f)) for h∈H,φ∈Ak(n(w)) and f∈Ak(n(w)).
A splitting φof Ak(n(w)) is called H-equivariant if h∗φ=φ, for all h∈H. It is
easy to see that a splitting φis H-equivariant if and only if
φAk(n(w))µ⊂Ak(n(w))µ/p for µ∈Q+.
Here Ak(n(w))µ/p is understood as zero space if µ∈ pQ+.
Proposition 16. The splitting φwis the unique H-equivariant splitting which
compatibly splits all the divisors Yi(i∈Supp(w)).
Moreover, this unique splitting is given by σp−1
wunder the isomorphism (25) up
to a k∗-scalar, where
σw= ( Y
i∈Supp(w)
pi)v−1∈H0(N(w), ω−1
N(w)).
Proof. Thanks to Proposition 15 and (10), the splitting φwsatisfies the require-
ment. We next prove the uniqueness.
Let φbe an H-equivariant Frobenius splitting which compatibly splits Yi, for
all i∈Supp(w). Suppose φis given by gv1−p∈H0(N(w), ω1−p
N(w)) for some g∈
Ak(n(w)). By [4, Lemma 4.1.14], the isomorphism (25) is moreover H-equivariant.
Since φis H-equivariant and v1−pis an H-eigenvector, we deduce that gis an
H-eigenvector. Equivalently, the element gis homogeneous with respect to the
Q+-grading.
Take a reduced expression w= (ir, . . . , i1) of w. Set βk=si1. . . sik−1(αik)∈Q+
for 1 ≤k≤r. Recall the elements yt(1 ≤t≤r) and the equality (23) in §3.2.
We write gin terms of the polynomial on variables y1, . . . , yr. Then the mono-
mial yp−1
1. . . yp−1
roccurs with a nonzero coefficient ([4, Example 1.3.1]). Hence
wt(g) = wt(yp−1
1. . . yp−1
r)=(p−1)(β1+· · · +βr).(26)
On the other hand, take any i∈Supp(w). We claim that pp−1
idivides g.
Suppose the claim is not correct. Write g=pl
ig1, with 0 ≤l < p −1 and
(pi, g1) = 1.
22 JINFENG SONG
Take x∈Yito be a smooth point of Yi. Then we can choose a system of local
coordinates t1, . . . , trat xwith t1=piin the local ring ON(w),x . Since g1(x)= 0,
the section g1v1−pdoes not vanish at x, which implies the local expression of g1v1−p
have nonzero constant term. Therefore at the point x, the section gv1−p=tl
1g1v1−p
has the local expression of the form
tl
1(a+X
j=0
djtj)(dt1∧ · · · ∧ dtr)1−pwith a= 0.
By [4, Lemma 1.3.6 & Proposition 1.3.7], the splitting φwinduces a splitting on
the local ring ON(w),x ⊂k[[t1, . . . , tr]] given by
fφw(f) = Trftl
1(a+X
j=0
djtj)for f∈ ON(w),x.
Here Tr is the trace map on the ring of formal power series defined as in [4, Lemma
1.3.6]. Moreover fφwshould preserves the principal ideal generated by t1, because
φwcompatibly splits Yi.
Note that the monomial tp−1−l
1tp−1
2. . . tp−1
rbelongs to t1ON(w),x, but its image
fφw(tp−1−l
1tp−1
2. . . tp−1
r)∈k[[t1, . . . , tr]]
has constant term a, which implies that it does not belong to t1ON(w),x. This is a
contradiction.
We have proved that pp−1
i|gfor any i∈Supp(w). Hence we can write
g=Y
i∈Supp(w)
pp−1
ig′,
for some g′∈Ak(n(w)).
Notice that
wtY
i∈Supp(w)
pp−1
i= (p−1)X
i∈Supp(w)
ϖi−w−1(X
i∈Supp(w)
ϖi)= (p−1)(β1+· · ·+βr),
where the last equality follows from [20, 1.3.22]. Combining (26) we deduce that
wt(g′) = 0. Hence, g′belongs to k∗.
Therefore φis unique and it is determined by σp−1
wup to a nonzero scalar. We
complete the proof. □
3.4. Comparison with geometric construction. From now on, we assume
that our root datum is of finite type.
Let Gbe the reductive group defined over kassociated to the root datum, with
a standard Borel subgroup Band a standard opposite Borel subgroup B−. Let N
and N−be the unipotent radicals of Band B−, respectively. Set H=B∩B−to
be the maximal torus, and W=N(H)/H to be the Weyl group. For any w∈W,
set
N(w) = N∩w−1N−w.
It is a closed subgroup of the unipotent group N. Let Hacts on N(w) by conju-
gation. Note that one has N=N(w0), where w0∈Wis the longest element in
the Weyl group.
QUANTUM FROBENIUS SPLITTINGS AND CLUSTER STRUCTURES 23
The coordinate ring of His canonically isomorphic to the group algebra k[P],
and the coordinate ring of the closed subgroup N∩w−1N−wis canonically isomor-
phic to Ak(n(w)) as k-algebras by [17, Theorem 4.44]. Moreover, the H-action on
N(w) by conjugation coincides with the one defined by the Q+-grading as in §3.3.
Therefore there is no conflict with previous notations.
We notice that although the result in [17] is over the field of complex numbers,
the proof applies to arbitrary fields, because the author essentially works over
integral forms.
Let G/B be the associated flag variety and BwB/B be the Schubert cell for
w∈W. Let Hacts on Schubert cells by left multiplications. Then there is an
H-equivariant isomorphism,
ιw:N(w)∼
−! Bw−1B/B, given by n7! n·w−1B/B. (27)
By [4, Theorem 4.1.15] the flag variety G/B admits a B-canonical (see [4, §4.1]
for the definition) Frobenius splitting φwhich compatibly splits Schubert varieties
BwB/B and opposite Schubert varieties B−wB/B, for any w∈W. Therefore
the splitting φinduces a splitting of the Schubert cell Bw−1B/B, which we call
the canonical splitting. Under the isomorphism ιw, we get a splitting φ′
wof N(w).
Corollary 17. As Frobenius splittings of N(w), we have φw=φ′
w.
Proof. For λ∈P+, set Vk(λ) = k⊗Z[q±1]V(λ). Then Vk(λ) carries a rational
G-action. The bilinear form k(·,·)λon Vk(λ) is defined to be the base change of
the form defined in §2.4. Then for any i∈Iand x∈N(w) we have
k(xvw−1ϖi, vϖi)ϖi= 0 ⇐⇒ k(x˙w−1vϖi, vϖi)ϖi= 0
⇐⇒ x˙w−1∈B−siB
⇐⇒ ιw(x)∈B−siB/B,
where ˙w∈Gis a lift of the Weyl group element w.
Therefore we have
ιw(Yi) = Bw−1B/B ∩B−siB/B, for i∈Supp(w). (28)
By [4, Lemma 1.1.7], the induced splitting of Bw−1B/B compatibly splits the
intersection Bw−1B/B ∩B−siB/B, so the splitting φ′
wcompatibly splits Yifor
any i∈Supp(w) by (28). Since φis B-canonical and ιwis H-equivariant, we
deduce that φ′
wis H-equivariant. Hence by Proposition 16 we have φw=φ′
w.□
3.5. Relation with reduction maps. In this subsection we explain that the
canonical splittings of Schubert cells are compatible with reduction maps.
Suppose w=v′vin Wwith l(w) = l(v′) + l(v). We choose a reduced expression
w= (il(w), . . . , i1) such that (il(v)...,i1) gives a reduced expression of v. It follows
from the definitions that we have a natural embedding as k-algebras,
Ak(n(v)) Ak(n(w)).
We prove Corollary 3 here. By (24), take isomrphisms N(w)∼
=Al(w)and
N(v)∼
=Al(v)associated to the chosen reduced expressions. It is direct to see that
under the isomorphisms ιwand ιv, the reduction map is just the projection onto the
24 JINFENG SONG
first l(v) coordinates. Therefore the comorphism (πw
v)∗:Ak(n(v)) !Ak(n(w))
coincides with the natural embedding.
Hence it will suffice to check the commutativity of the following diagram as
abelian groups,
Ak(n(v)) Ak(n(w))
Ak(n(v)) Ak(n(w)).
φvφw
This diagram clearly commutes, because by construction φvand φware both
restrictions of the same map on Ak(n).
Declarations
Ethical Approval
not applicable
Funding
The author is supported by Huanchen Bao’s MOE grant A-0004586-00-00 and
A-0004586-01-00.
Availability of data and materials
not applicable
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Department of Mathematics, National University of Singapore, Singapore.
Email address:j song@u.nus.edu
