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Quantum Frobenius Splittings and Cluster Structures

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Jinfeng Song at National University of Singapore
Jinfeng Song
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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.
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Algebras and Representation Theory (2024) 27:1773–1797
https://doi.org/10.1007/s10468-024-10281-x
RESEARCH
Quantum Frobenius Splittings and Cluster Structures
Jinfeng Song1
Received: 12 November 2023 / Accepted: 23 July 2024 / Published online: 5 August 2024
© The Author(s), under exclusive licence to Springer Nature B.V. 2024
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.
Keywords Frobenius splittings ·Cluster algebras ·Quantum groups
Mathematics Subject Classification (2010) 13F60 ·17B37
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,andεbe 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
afew,themapFr plays essential role in the study of representations of reductive groups over
positive characteristic [1,14], and the map Fris used to construct Frobenius splittings of
Schubert varieties in [21].
Lusztig’s original definitions of these two maps are only derived through brute force com-
putations. 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 con-
struction of the map Fr.Qi[25] categorified Fr for g=sl2at prime roots of unity via p-DG
algebras. The map Frseems 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 Frsimultaneously.
Presented by: Peter Littelmann
BJinfeng Song
j_song@u.nus.edu
1Department of Mathematics, National University of Singapore, 21 Lower Kent Ridge Road, 119077
Kent Ridge, Singapore
123
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THE DRINFELD JIMBO ALGERBRA U.- The Algebra f.- Weyl Group, Root Datum.- The Algebra U.- The Quasi--Matrix.- The Symmetries of an Integrable U-Module.- Complete Reducibility Theorems.- Higher Order Quantum Serre Relations.- GEOMETRIC REALIZATION OF F.- Review of the Theory of Perverse Sheaves.- Quivers and Perverse Sheaves.- Fourier-Deligne Transform.- Periodic Functors.- Quivers with Automorphisms.- The Algebras and k.- The Signed Basis of f.- KASHIWARAS OPERATIONS AND APPLICATIONS.- The Algebra .- Kashiwara's Operators in Rank 1.- Applications.- Study of the Operators .- Inner Product on .- Bases at ?.- Cartan Data of Finite Type.- Positivity of the Action of Fi, Ei in the Simply-Laced Case.- CANONICAL BASIS OF U.- The Algebra .- Canonical Bases in Certain Tensor Products.- The Canonical Basis .- Inner Product on .- Based Modules.- Bases for Coinvariants and Cyclic Permutations.- A Refinement of the Peter-Weyl Theorem.- The Canonical Topological Basis of .- CHANGE OF RINGS.- The Algebra .- Commutativity Isomorphism.- Relation with Kac-Moody Lie Algebras.- Gaussian Binomial Coefficients at Roots of 1.- The Quantum Frobenius Homomorphism.- The Algebras .- BRAID GROUP ACTION.- The Symmetries of U.- Symmetries and Inner Product on f.- Braid Group Relations.- Symmetries and U+.- Integrality Properties of the Symmetries.- The ADE Case.
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