Epsilon Systems on Geometric Crystals of Type $A_n$

We introduce an epsilon system on a geometric crystal of type $A_n$, which is a certain set of rational functions with some nice properties. We shall show that it is equipped with a product structure and that it is invariant under the action of tropical R maps.


Introduction
In the theory of crystal bases, the piecewise-linear functions ε i and ϕ i play many crucial roles, e.g., description of highest weight vectors, tensor product of crystals, extremal vectors, etc. There exist counterparts for geometric crystals, denoted also {ε i }, which are rational functions with several nice properties, indeed, they are needed to describe the product structure of geometric crystals (see Section 2) and in sl 2 -case, the universal tropical R map is presented by using them [12].
In [1], higher objects ε i,j and ε j,i are introduced in order to prove the existence of product structure of geometric crystals induced from unipotent crystals, which satisfy the relation ε i ε j = ε i,j + ε j,i if the vertices i and j are simply laced. It motivates us to define further higher objects, "epsilon system".
The aim of the article is to define an "epsilon system" for type A n and reveal its basic properties, e.g., product structures and invariance under the action of tropical R maps. An epsilon system is a certain set of rational functions on a geometric crystal, which satisfy some relations with each other and have simple forms of the action by e c i 's. We found its prototype on the geometric crystal of the opposite Borel subgroup B − ⊂ SL n+1 (C). In that case, indeed, the epsilon system is realized as a set of matrix elements and minor determinants of unipotent part of a group element in B − (see Section 6). Therefore, we know that geometric crystals induced from unipotent crystals are equipped with an epsilon system naturally.
We shall introduce two remarkable properties of epsilon system: One is a product structure of epsilon systems. That is, for two geometric crystals with epsilon systems, say X and Y, there exists canonically an epsilon system on the product of geometric crystals X × Y (see Section 5.3).
In the last section, we shall give an application of these results, which shows the uniqueness of tropical R map on some geometric crystals of type A (1) n . Since the sl 2 -universal tropical R map is presented by using the rational functions {ε i } as mentioned above, we expect that epsilon systems would be a key to find universal tropical R maps of higher ranks. For further aim, we would like to extend this notion to other simple Lie algebras, e.g., B n , C n , and D n . These problems will be discussed elsewhere.

Geometric crystals
Fix a symmetrizable generalized Cartan matrix A = (a ij ) i,j∈I with a finite index set I. Let (t, {α i } i∈I , {h i } i∈I ) be the associated root data satisfying α j (h i ) = a ij . Let g = g(A) = t, e i , f i (i ∈ I) be the Kac-Moody Lie algebra associated with A [5]. Let P ⊂ t * (resp. Q := ⊕ i Zα i , Q ∨ := ⊕ i Zh i ) be a weight (resp. root, coroot) lattice such that C ⊗ P = t * and P ⊂ {λ | λ(Q ∨ ) ⊂ Z}, whose element is called a weight. Define the simple reflections s i ∈ Aut(t) (i ∈ I) by s i (h) := h − α i (h)h i , which generate the Weyl group W . Let G be the Kac-Moody group associated with (g, P ) [6,7]. Let U α := exp g α (α ∈ ∆ re ) be the one-parameter subgroup of G. The group G (resp. U ± ) is generated by Here U ± is a unipotent subgroup of G. For any i ∈ I, there exists a unique group homomorphism φ i : The relations in (iii) is called Verma relations. If χ = (X, {e i }, {γ i }, {ε i }) satisfies the conditions (i), (ii) and (iv), we call χ a pre-geometric crystal.
Remark. The last condition (iv) is slightly modified from [2,9,10,11,12] since all ε i appearing in these references satisfy the new condition 'ε i (e c j (x)) = ε i (x) if a i,j = a j,i = 0' and we need this condition to define "epsilon systems" later.

Unipotent crystals
In the sequel, we denote the unipotent subgroup U + by U . We define unipotent crystals (see [1,9]) associated to Kac-Moody groups.
Definition 2.2. Let X be an ind-variety over C and α : U × X → X be a rational U -action such that α is defined on {e} × X. Then, the pair X = (X, α) is called a U -variety. For U -varieties X = (X, α X ) and Y = (Y, α Y ), a rational map f : X → Y is called a U -morphism if it commutes with the action of U . Now, we define a U -variety structure on B − = U − T . As in [8], the Borel subgroup B − is an ind-subgroup of G and hence an ind-variety over C. The multiplication map in G induces the open embedding; B − × U ֒→ G, which is a birational map. Let us denote the inverse birational map by g : G −→ B − ×U and let rational maps π − : G → B − and π : G → U be π − := proj B − •g and π := proj U • g. Now we define the rational U -action α B − on B − by  (i) Let X = (X, α) be a U -variety and f : X → B − a U -morphism. The pair (X, f ) is called a unipotent G-crystal or, for short, unipotent crystal.
In particular, if g is a birational map of ind-varieties, it is called an isomorphism of unipotent crystals.
We define a product of unipotent crystals following [1]. For unipotent crystals (X, f X ), Then f X×Y is a U -morphism and (X × Y, f X×Y ) is a unipotent crystal, which we call a product of unipotent crystals (X, f X ) and (Y, f Y ).
(iii) Product of unipotent crystals is associative.

From unipotent crystals to geometric crystals
where x α i (t) := x i (t) and x −α i (t) := y i (t). We have the unique decomposition; By using this decomposition, we get the canonical projection ξ i : U − → U −α i . Now, we define the function on U − by and extend this to the function on B − by χ i (u·t) := χ i (u) for u ∈ U − and t ∈ T . For a unipotent G-crystal (X, f X ), we define a function ε i := ε X i : X → C by and a rational function γ i : X → C by where proj T is the canonical projection.
Remark. Note that the function ε i is denoted by ϕ i in [1,9].
Suppose that the function ε i is not identically zero on X. We define a morphism e i : Theorem 2.5 ([1]). For a unipotent G-crystal (X, f X ), suppose that the function ε i is not identically zero for any i ∈ I. Then the rational functions γ i , ε i : X → C and e i : Then we obtain: .
(ii) For any i ∈ I, the action e Z i : , .
Here note that c 1 c 2 = c. The formula c 1 and c 2 in [1] seem to be different from ours.

Prehomogeneous geometric crystal
be a geometric crystal. We say that χ is prehomogeneous if there exists a Zariski open dense subset Ω ⊂ X which is an orbit by the actions of the e c i 's.
be a finite-dimensional positive geometric crystal with the positive structure θ : (C × ) dim(X) → X and B := U D θ (χ) the crystal obtained as the ultra-discretization of χ. If B is a connected crystal, then χ is prehomogeneous.
In [2,3], we showed that ultra-discretization of the affine geometric crystal V(g) l (l > 0) is a limit of perfect crystal B ∞ (g L ), where g L is the Langlands dual of g. Since for any k ∈ Z >0 a tensor product B ∞ (g L ) ⊗k is connected by the perfectness of B ∞ (g L ) and we have the isomorphism of crystals by Theorem 3.3 we obtain the following:

Tropical R maps
Definition 4.1. Let {X λ } λ∈Λ be a family of geometric crystals with the product structures, where Λ is an index set. A birational map R λµ : for any i ∈ I and λ, µ, ν ∈ Λ.
Tropical R maps for certain affine geometric crystals of type A 2n are described explicitly [3,4]. The following is immediate from Lemma 3.2 and Corollary 3.4.
Let us introduce an example of a tropical R map of type A  . . , l n+1 ) | l 1 l 2 · · · l n+1 = L}, which is equipped with an A (1) n -geometric crystal structure by: The tropical R map on {B L } L∈R >0 is given by [4]: Remark. In the case g = A For a partition P = {I 1 , . . . , I k } and symbols ε I j (j = 1, . . . , k), define , the set of rational functions on X, say E = {ε J , ε * J |J ∈ J is an interval }, is called an epsilon system of X if they satisfy the following: for any interval J = {s, s + 1, . . . , t} ⊂ I, and we set ε J := ε i for J = {i}, which is originally equipped with X. Note that ε * i = ε i . We call a geometric crystal with an epsilon system an ε-geometric crystal.
The actions of e c s−1 and e c t+1 will be described explicitly below, which are derived from (5.1) and (5.2).
Proposition 5.3. The above definition is well-defined, that is, for any J ∈ J we have More precisely, we claim that if we calculate the both sides of the above equations by using (5.1) and (5.2), they coincide with each other.
The proof will be given in the next subsection.
for any s, t ∈ I such that s ≤ t.
Proof . The proof is easily done by using the induction on t − s.
The following describes the explicit action of e c i on epsilon systems which is not given in Definition 5.2.
Proposition 5.6. We have the following formula for s ≤ t: . (5.10) Proof . Let us only show (5.7) since the others are shown similarly. It follows from (5.5) for [s, t + 1] that Then applying e c t to x, we get By this result, we know that all explicit forms of the action by e c i on epsilon systems. .
Thus, we obtained (5.11). The others are also shown by direct calculations: Let g be a Kac-Moody Lie algebra associated with the index set I and g J be a subalgebra associated with a subset J ⊂ I. Let X = (X, {γ i }, {ε i }, {e i }) i∈I be a g-geometric crystal. Then it has naturally a g J -geometric crystal structure and denote it by X J .
Definition 5.7. In the above setting, if g J is isomorphic to the Lie algebra of type A n for some n and the geometric crystal X J has an epsilon system E X J of type A n , then we call it a local epsilon system of type A n associated with the index set J.
Remark. An ε-geometric crystal has naturally a local epsilon system associated with each sub-interval of I.

Product structures on epsilon systems
Theorem 5.9. Let X and Y be ε-geometric crystals. Suppose that the product X × Y has a geometric crystal structure. Then X × Y turns out to be an ε-geometric crystal as follows: for ε-systems E X := {ε X J , ε X * J } J∈J of X and . (5.16) ∈J defines an epsilon system of X × Y. Example 5.10. We have .
Thus, by the induction on t − s, we obtain (5.2).
Next, we shall see an epsilon system on a geometric crystal induced from a unipotent crystal. Let (X, f ) be a unipotent SL n+1 (C) crystal, where f : X → B − is a U -morphism. We assume that any rational function ε i is not identically zero. By Theorem 2.5, we get the geometric crystal X = (X, {e c i }, {γ i }, {ε i }).
Theorem 6.2. The geometric crystal X as above is an ε-geometric crystal. Indeed, by setting the set E X := {ε X [s,t] (x), ε * X [s,t] (x)|1 ≤ s ≤ t ≤ n} defines an epsilon system on X.
Remark. We expect that this method is applicable to the tropical R maps of other types [3]. But we do not have explicit answers.