Geometric Lifting of Affine Type A Crystal Combinatorics by Gabriel Frieden A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in the University of Michigan 2018 Doctoral Committee: Professor Thomas Lam, Chair Associate Professor Victoria Booth Professor Sergey Fomin Professor William Fulton Professor David E Speyer Gabriel Frieden [email protected] ORCID iD: 0000-0001-6231-9365 (cid:13)c Gabriel Frieden 2018 ACKNOWLEDGMENTS I would like to start by thanking my advisor Thomas Lam for igniting my interest in algebraic combinatorics with his course on total positivity, encouraging me to work on the problem of lifting the combinatorial R-matrix, and guiding me through the research process. I came away from each of the many hours at his blackboard with specific ideas to try, as well as a better feeling for the big picture. I am also grateful to Thomas for generously supporting my travel to conferences, which played an important role in my mathematical development. I am grateful to John Stembridge for his beautiful (and crystal-clear) courses on Coxeter groups and combinatorial representation theory; Sergey Fomin for his wonderful course on cluster algebras, and for carefully reading a draft of this thesis; David Speyer for pointing me toward the existing literature on geometric lifting of tableaucombinatoricsatthebeginningofthisproject; BillFultonforshowingmehow an algebraic geometer thinks; and Karen Smith for her helpful advice throughout my time in graduate school. Thanks also to the many graduate students and post-docs who have made the last several years fun, both mathematically and otherwise. I would like to thank my parents Susan and Karl for their loving support, and for encouraging me to pursue my interest in mathematics; my sisters Eva and Kyla for their unique perspectives on everything from food to sports to the arts; and my grandfather David Buchsbaum for his enthusiasm for mathematics in general, and especially whatever I have recently learned in my courses or research. Last but certainly not least, thanks to Leigh for patiently listening to me talk about all things mathematical, and for being the best companion I could have asked for through the ups and downs of the past several years. Work on this thesis was partially supported by NSF grants DMS-1464693 and DMS-0943832. ii TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . ii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi CHAPTER 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Affine crystals and the combinatorial R-matrix . . . . . . . . 1 1.2 Geometric lifting . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Cyclic symmetry and the Grassmannian . . . . . . . . . . . . 6 1.4 Unipotent crystals and the loop group . . . . . . . . . . . . . 8 1.5 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6 Applications and future directions . . . . . . . . . . . . . . . 9 1.7 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Crystal axioms . . . . . . . . . . . . . . . . . . . . . 14 2.2.2 Crystal structure on tableaux . . . . . . . . . . . . 15 2.2.3 Tensor product of crystals . . . . . . . . . . . . . . 19 2.2.4 Piecewise-linear translation . . . . . . . . . . . . . . 22 2.3 Geometric crystals . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Unipotent crystals . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 The Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6 Planar networks and the Lindstro¨m Lemma . . . . . . . . . . 40 iii 3. Geometric and unipotent crystals on the Grassmannian . . . 42 3.1 Main definitions . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 Properties of the matrix g(N|t) . . . . . . . . . . . . . . . . . 48 3.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.1 Z/nZ symmetry . . . . . . . . . . . . . . . . . . . . 50 3.3.2 The geometric Schu¨tzenberger involution . . . . . . 52 3.3.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . 58 4. From geometry to combinatorics . . . . . . . . . . . . . . . . . . 64 4.1 The Gelfand–Tsetlin parametrization . . . . . . . . . . . . . . 64 4.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . 64 4.1.2 Network representation and formulas for Plu¨cker co- ordinates . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Positivity and tropicalization . . . . . . . . . . . . . . . . . . 72 4.2.1 Positive varieties and positive rational maps . . . . 72 4.2.2 Definition of tropicalization . . . . . . . . . . . . . . 76 4.3 Recovering the combinatorial crystals . . . . . . . . . . . . . 77 4.3.1 Tropicalizing the geometric crystal maps and sym- metries . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3.3 Proof of Lemma 4.22 . . . . . . . . . . . . . . . . . 83 4.3.4 Proof of Theorem 4.24 . . . . . . . . . . . . . . . . 84 5. Lifting the combinatorial R-matrix . . . . . . . . . . . . . . . . 89 5.1 The geometric R-matrix . . . . . . . . . . . . . . . . . . . . . 89 5.1.1 Definition of R . . . . . . . . . . . . . . . . . . . . . 89 5.1.2 Properties of R . . . . . . . . . . . . . . . . . . . . 91 5.1.3 Recovering the combinatorial R-matrix . . . . . . . 93 5.2 The geometric coenergy function . . . . . . . . . . . . . . . . 94 5.3 One-row tableaux . . . . . . . . . . . . . . . . . . . . . . . . 99 5.4 Proof of the positivity of the geometric R-matrix . . . . . . . 105 5.5 Proof of the identity g ◦R = g . . . . . . . . . . . . . . . . . 112 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 iv LIST OF FIGURES Figure 1 The Kirillov–Reshetikhin crystal B2,2 of type A(1). . . . . . . . . . . . . . . 16 2 2 An example of r(cid:102)efl in the L=1 case (with n=7,k =4). Here b corresponds to the partition (3,2,2,1) and the subset {1,3,4,6}; r(cid:102)efl(b) corresponds to (3,1,0) and {2,5,7}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 A planar network and its associated matrix. Unlabeled edges have weight 1. . . 41 4 The network Γ . Unlabeled edges have weight 1. . . . . . . . . . . . . . . 67 2,5 5 A vertex-disjoint family of paths in Γ that contributes to the Plu¨cker coordi- 5,9 nate P , and the corresponding {1,3,4,6,8}-tableau. . . . . . . . . . . . 68 13468 6 Anetworkrepresentationofthematrixg(Θ (X)). Verticaledgeshaveweight1.100 n−1 7 Suppose n=4,(cid:96)=k =2, and u=M|s,v =N|t∈X . The first line shows the 2 product g(u)g(v) = A, where the ∗’s are ratios of Plu¨cker coordinates of M or N, possibly scaled by s or t, and X is the unfolding of A. The next lines show the product g(u)∗g(v)∗ =A∗, with the blocks of A∗ indicated. . . . . . . . . 115 8 The matrix A∗ in the case n=7,(cid:96)=4,k =2, with blocks indicated. . . . . . 116 v ABSTRACT In the first part of this thesis, we construct a type A(1) geometric crystal on the n−1 variety X := Gr(k,n) × C×, and show that it tropicalizes to the disjoint union of k the Kirillov–Reshetikhin crystals corresponding to rectangular semistandard Young tableauxwithn−k rows. AkeyingredientinourconstructionistheZ/nZsymmetry of the Grassmannian which comes from cyclically shifting a basis of the underlying vector space. We show that a twisted version of this symmetry tropicalizes to com- binatorial promotion. In the second part, we define and study the geometric R-matrix, a birational map R : X ×X → X ×X which tropicalizes to the combinatorial R-matrix on pairs k1 k2 k2 k1 of rectangular tableaux. We show that R is an isomorphism of geometric crystals, and that it satisfies the Yang–Baxter relation. In the case where both tableaux have one row, we recover the birational R-matrix of Yamada and Lam–Pylyavskyy. Most of the properties of the geometric R-matrix follow from the fact that it gives the unique solution to a certain equation of matrices in the loop group GL (C(λ)). n vi CHAPTER 1 Introduction 1.1 Affine crystals and the combinatorial R-matrix In the early 1990s, Kashiwara introduced the theory of crystal bases [Kas90, Kas91]. This groundbreaking work provides a combinatorial model for the represen- tation theory of semisimple (and more generally, Kac–Moody) Lie algebras, allowing many aspects of the representation theory to be studied from a purely combinatorial point of view. In type A, crystal bases can be realized as a collection of combinatorial maps on semistandard Young tableaux, and many previously studied combinatorial tableau algorithms turned out to be special cases of crystal theory. For example, the Robinson–Schensted–Knuth correspondence is the crystal version of the decom- position of the GL -representation (Cn)⊗d into its irreducible components [Shi05]; n Lascoux and Schu¨tzenberger’s symmetric group action on tableaux is a special case of the Weyl group action on any crystal [BS17]; Schu¨tzenberger’s promotion map, re- stricted to rectangular tableaux, is the crystal-theoretic manifestation of the rotation of the affine type A Dynkin diagram [Shi02]. Tableaualgorithmsaretraditionallydescribedasasequenceoflocalmodifications to a tableau, such as bumping an entry from one row to the next, or sliding an entry into an adjacent box. These combinatorial descriptions are quite beautiful, but for some purposes, one might want a formula that describes the local transformations in terms of a natural set of coordinates on tableaux, such as the number of j’s in the ith row (or the closely related Gelfand–Tsetlin patterns). Kirillov and Berenstein discovered that the Bender–Knuth involutions, which are the building blocks for algorithms such as promotion and evacuation, act on a Gelfand–Tsetlin pattern by simple piecewise-linear transformations [KB96]. This discovery sparked a search for piecewise-linear formulas for other combinatorial algorithms. This thesis is centered around the problem of finding piecewise-linear formulas for combinatorial maps coming from affine crystal theory. Quantum affine algebras ad- mit a class of finite-dimensional, non-highest-weight representations called Kirillov– 1 2 Reshetikhin (KR) modules. The crystal bases of these representations, which we call KR crystals, have received a lot of attention for several reasons. Kang et al. showed that the crystal bases of highest-weight modules for quantum affine algebras can be built out of infinite tensor products of KR crystals, and they used this construc- tion to compute the 1 point functions of certain solvable lattice models coming from statistical mechanics [KKM+92]. Kirillov–Reshetikhin crystals have also played a central role in the study of a cellular automaton called the box-ball system and its generalizations [TS90, HHI+01]. Unlike the tensor product of representations of Lie algebras and finite groups, the tensor product of representations of quantum algebras (and thus of crystals) is not commutative. In the case of KR crystals, however, there is a unique crystal isomorphism R(cid:101) : B ⊗B → B ⊗B . 1 2 2 1 This isomorphism is called the combinatorial R-matrix, and it plays an essential role in both of the applications mentioned in the preceding paragraph. For example, the states of the box-ball system can be represented as elements of a tensor product of KR crystals, and the time evolution is given by applying a sequence of combinatorial R-matrices. In (untwisted) affine type A, Kirillov–Reshetikhin modules correspond to parti- tions of rectangular shape (Lk), and their crystal bases, which we denote by Bk,L, are modeled by semistandard Young tableaux of shape (Lk). If one ignores the affine crystal operators e ,f(cid:101), then Bk,L is the crystal associated to the irreducible sl - (cid:101)0 0 n module of highest weight (Lk). Shimozono showed that the affine crystal operators are obtained by conjugating the crystal operators e ,f(cid:101) by Schu¨tzenberger’s pro- (cid:101)1 1 motion map [Shi02]. He also gave a combinatorial description of the action of the combinatorial R-matrix on pairs of rectangular tableaux, which we now explain. Let ∗ denote the associative product on the set of semistandard Young tableaux introduced by Lascoux and Schu¨tzenberger (see §2.2.3 for the definition). If T ∈ Bk1,L1 and U ∈ Bk2,L2, then there are unique tableaux U(cid:48) ∈ Bk2,L2 and T(cid:48) ∈ Bk1,L1 such that T ∗ U = U(cid:48) ∗ T(cid:48), and the combinatorial R-matrix is realized by the map R(cid:101) : T ⊗U (cid:55)→ U(cid:48) ⊗T(cid:48). For example, suppose T = 1 2 2 3 4 4 4 5 ∈ B1,8 and U = 1 2 2 4 5 5 ∈ B1,6. The product T ∗ U can be computed by using Schensted’s row bumping algorithm to insert the entries of U into T, starting from the left end of U; the result is 1 1 2 2 2 4 4 4 5 5 T ∗U = . 2 3 4 5 3 The reader may verify that the tableaux U(cid:48) = 1 2 2 3 4 5 and T(cid:48) = 1 2 2 4 4 4 5 5 satisfy U(cid:48) ∗T(cid:48) = T ∗U, so R(cid:101)(T ⊗U) = U(cid:48) ⊗T(cid:48). There is a combinatorial procedure for pulling T ∗ U apart into U(cid:48) and T(cid:48), so the whole process is algorithmic. It is nevertheless natural to ask if the map R(cid:101) can be computed in one step, without first passing through the product T ∗ U. In the case where T and U are both one-row tableaux, there is an elegant piecewise-linear formula for R(cid:101), due to Hatayama et al. Proposition 1.1 ([HHI+01, Prop. 4.1]). Suppose T and U are one-row tableaux, with entries at most n, and suppose R(cid:101)(T ⊗U) = U(cid:48) ⊗T(cid:48). Let a ,b be the numbers j j of j’s in T and U, respectively. Define b(cid:48) = b +κ −κ , a(cid:48) = a +κ −κ , j j (cid:101)j+1 (cid:101)j j j (cid:101)j (cid:101)j+1 where κ = min (b +b +···+b +a +a +···+a ), (cid:101)j j j+1 j+r−1 j+r+1 j+r+2 j+n−1 0≤r≤n−1 and all subscripts are interpreted modulo n. Then b(cid:48),a(cid:48) are the numbers of j’s in U(cid:48) j j and T(cid:48), respectively. The motivating goal of this thesis was to generalize Proposition 1.1 to a formula for the combinatorial R-matrix on pairs of arbitrary rectangular tableaux. 1.2 Geometric lifting How does one find—and work with—piecewise-linear formulas for complicated combinatorial operations? A very useful method is to use tropicalization and geo- metric lifting. Tropicalization is the procedure which turns a positive rational func- tion (i.e., a function consisting of the operations +,·,÷, but not −; such functions are often called “subtraction-free” in the literature) into a piecewise-linear function by making the substitutions (+,·,÷) (cid:55)→ (min,+,−). A geometric (or rational) lift of a piecewise-linear function (cid:101)h is any positive rational function h which tropicalizes to (cid:101)h. Rational functions are often easier to work with than piecewise-linear functions, since one may bring to bear algebraic and geomet- ric techniques. Furthermore, identities proved in the lifted setting can be “pushed