The U(n) Gelfand–Zeitlin system as a tropical limit of Ginzburg–Weinstein diffeomorphisms

Abstract

In this paper, we show that the Ginzburg–Weinstein diffeomorphism of Alekseev & Meinrenken (Alekseev, Meinrenken 2007 J. Differential Geom. 76, 1–34. (10.4310/jdg/1180135664)) admits a scaling tropical limit on an open dense subset of . The target of the limit map is a product , where is the interior of a cone, T is a torus, and carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to recovers the Gelfand–Zeitlin integrable system of Guillemin & Sternberg (Guillemin, Sternberg 1983 J. Funct. Anal. 52, 106–128. (10.1016/0022-1236(83)90092-7)). As a by-product of our proof, we show that the Lagrangian tori of the Flaschka–Ratiu integrable system on the set of upper triangular matrices meet the set of totally positive matrices for sufficiently large action coordinates.

This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

1. Introduction

One of the richest settings for the study of integrable systems is the dual vector space of a finite dimensional Lie algebra , equipped with its canonical Lie–Poisson structure. There are many important examples of integrable systems defined on , including spinning tops [1] and Mishchenko–Fomenko systems [2,3]. Systems on also give rise to collective integrable systems via moment maps [4,5], for instance, leading to complete integrability of the geodesic flow on certain homogeneous spaces [6].

Gelfand–Zeitlin systems, defined by Guillemin and Sternberg on the space of Hermitian matrices (interpreted as ), are one of the most famous examples of such integrable systems [7]. Unlike Mishchenko–Fomenko systems, Gelfand–Zeitlin systems have natural global action-angle coordinates. This structure has led to results about the symplectic topology of coadjoint orbits [8,9]. Unfortunately, Gelfand–Zeitlin systems have only been defined for of types A, B and D.

Motivated by the problem of generalizing Gelfand–Zeitlin systems, we are naturally brought to the following question: What underlying structures give rise to Gelfand–Zeitlin systems? One answer to this question comes from the study of toric degenerations, which define integrable systems with global action-angle coordinates on coadjoint orbits and many other spaces [9,10]. In this paper, we make progress towards a new answer to this question by relating Gelfand–Zeitlin systems on to Ginzburg–Weinstein diffeomorphisms and upper cluster algebra structures on dual Poisson–Lie groups. We believe that this approach can be generalized and will give examples of new integrable systems on Lie algebra duals that have natural global action-angle coordinates.

In more detail, let K be a compact connected Poisson–Lie group (that is, a Lie group equipped with a multiplicative Poisson bracket). Poisson–Lie theory associates with K a dual Poisson–Lie group K* which is solvable. The Ginzburg–Weinstein theorem says that K* admits global linearization maps: Poisson isomorphisms called Ginzburg–Weinstein diffeomorphisms.

When K = U(n), it is possible to describe explicit Ginzburg–Weinstein diffeomorphisms [11]. The dual Poisson–Lie group U(n)* has a completely integrable system with global action-angle coordinates due to Flaschka–Ratiu [12]. On open dense subsets of and U(n)*, a Poisson isomorphism is given by the identity in global action-angle coordinates for the Gelfand–Zeitlin and Flaschka–Ratiu systems (the main result of [11] is that this isomorphism extends to all of and U(n)*).

Returning to the general case of compact connected K, equip with its Lie–Poisson structure , the dual Poisson–Lie group K* with its Poisson structure π_K*, and fix a Ginzburg–Weinstein diffeomorphism . As is linear, for positive t the scaled Ginzburg–Weinstein diffeomorphism gw_t(A) = gw(tA) is a Poisson isomorphism with respect to and the scaled Poisson structure tπ_K*.

It was recently shown that integrable systems can be constructed from K* by tropicalizing its Poisson structure, which means taking the scaling limit t → ∞ of tπ_K* [13,14]. More precisely, denoting n = rank(K) and , there exist t-dependent coordinate charts (coming from an upper cluster algebra structure on a double Bruhat cell)

such that as t → ∞ the Poisson structure (Δ_t)_*(tπ_K*) converges to a constant Poisson structure π_∞, of rank m, on , where is the interior of a certain convex polyhedral cone and T^m is a torus. In terms of coordinates ζ on and φ on T^m, the constant Poisson structure π_∞ is of the form

where π_i,j are some constants (see theorem 3.1 and the following discussion and [13] for more details in the U(n) case. For the general case, see [14]). As π_∞ has rank m, the coordinates ζ are global action coordinates for an integrable system on (the coordinates φ differ from global angle coordinates by a linear transformation).

In general, we see that the composition

1.1

is a Poisson isomorphism with respect to and (Δ_t)_*(tπ_K*) for all positive finite t (on the open dense subset where the composition is defined). One may then make the following conjecture.

Conjecture 1.1 —

The limit as t → ∞ of the map (1.1) exists on an open dense subset and defines a Poisson isomorphism between , equipped with , and , equipped with π_∞.

If this conjecture is true, then the limit defines a completely integrable system with global action-angle coordinates on an open dense subset of . Our main result is that conjecture 1.1 holds for K = U(n).

Theorem 1.2 —

For the Ginzburg–Weinstein diffeomorphism of [11] and coordinate charts Δ_t defined by the cluster coordinates used in [13], conjecture 1.1 is true. Moreover, the integrable system on the open dense subset obtained by pulling back action-angle coordinates from is the Gelfand–Zeitlin system (up to a linear transformation).

Theorem 1.2 is illustrated by the following example of U(2) where the map gw_t can be made completely explicit. For U(n), n≥3, the map gw_t does not admit a tractable closed formula in standard matrix coordinates.

Example 1.3 —

If is identified with Hermitian 2 × 2 matrices, with coordinates

and U(2)* is identified with upper triangular matrices, with positive diagonal entries,

then the t-scaling of the Ginzburg–Weinstein diffeomorphism of [11] is given by the formula

where , z = ρ e^iθ, and t > 0. The action coordinates of the Gelfand–Zeitlin system are

and they satisfy ‘interlacing inequalities’ λ⁽²⁾₁≥λ⁽¹⁾₁≥λ⁽²⁾₂. The functions λ⁽²⁾₁, λ⁽²⁾₂ are Casimir functions and the function λ⁽¹⁾₁ generates a S¹ action on the coadjoint orbits (which are 2-spheres).

In coordinates (ζ⁽¹⁾₁, ζ⁽²⁾₁, ζ⁽²⁾₂, φ⁽²⁾₁) on , the chart Δ_t is given by

The open dense subset of is the set where the interlacing inequalities are strict. For A in this subset, one computes using the interlacing inequalities that

while ζ⁽¹⁾₁○gw_t = λ⁽¹⁾₁/2 and ζ⁽²⁾₂○gw_t = (λ⁽²⁾₁ + λ⁽²⁾₂)/2 for all t > 0. Thus,

The organization of this paper is as follows. In §2, we recall the definition of action-angle coordinates for the Gelfand–Zeitlin system on , some background from the theory of Poisson–Lie groups, and the explicit formula for the Ginzburg–Weinstein diffeomorphisms of [11]. Section 3 describes the cluster coordinates on U(n)* that were used in [13] to tropicalize the Poisson bracket on U(n)*, and explains their relation to the Gelfand–Zeitlin functions via the maps gw_t. In §4, we introduce matrix factorization coordinates on U(n)* and the tropical estimation results from [15], which are the main ingredients for proving convergence. Finally, §§5 and 6 are dedicated to the proof of theorem 1.2, which is divided into two propositions. Convergence of action coordinates is proved in proposition 5.1, and convergence of angle coordinates is proved in proposition 6.1. Part of the proof of proposition 6.1 involves showing that the Flaschka–Ratiu tori meet the set of totally positive matrices for sufficiently large values of action variables. A list of notation used throughout the paper is provided in table ??.

Table 1.

List of notation.

	The set of n × n Hermitian matrices
	The set of positive definite n × n Hermitian matrices
	The subset of where all interlacing inequalities are strict
λ(k)i	Gelfand–Zeitlin function on
ψ(k)i	Angle coordinates for the Gelfand–Zeitlin system on
ℓ(k)i	The sum λ(k)1 +  · s + λ(k)i
L	The Gelfand–Zeitlin map with coordinates ℓ(k)i
ΔI,J	Minor consisting of rows and columns I, J⊆{1, …, n}
Δ(k)i	Minor ΔI,J with I = {n − k + 1, …, n − k + i} and J = {n − i, …, n}
ζ(k)i, φ(k)i	Defined by
Δt	The t-dependent coordinate chart defined by ζ(k)i, φ(k)i
m(k)i	Tropical Gelfand–Zeitlin functions obtained by tropicalizing the polynomials (4.1)
	The tropical Gelfand–Zeitlin map with coordinates m(k)i
	The t-dependent Gelfand–Zeitlin map on AN
	The interior of the cone defined by rhombus inequalities (2.4)
	The subset of defined by inequalities (3.1)

2. Gelfand–Zeitlin and Ginzburg–Weinstein

In the first part of this section, we recall the definition of the classical Gelfand–Zeitlin system, along with several details about action-angle coordinates and the Gelfand–Zeitlin cone. In the second part of this section, we briefly recall the theory of Poisson–Lie groups, the Ginzburg–Weinstein theorem, and the explicit Ginzburg–Weinstein diffeomorphism of [11].

(a). The classical Gelfand–Zeitlin system

For any Lie group K with Lie algebra , the dual vector space is endowed with a linear Poisson bracket called the Lie–Poisson structure, defined by the formula

for and smooth functions .

Let K = U(n), the group of unitary n × n matrices, and let denote the set of Hermitian n × n matrices. We can identify via the non-degenerate bilinear form (X, Y ) = tr(XY ). With this identification, the Gelfand–Zeitlin functions on are the functions , 1 ≤ i ≤ k ≤ n, defined so that for ,

are the ordered eigenvalues of the k × k principal submatrix A^(k) in the bottom-right corner of A (figure 1).

Figure 1.

Bottom-right principal submatrices of A.

The Gelfand–Zeitlin functions satisfy ‘interlacing inequalities’,

2.1

and the image of the Gelfand–Zeitlin map , defined by the Gelfand–Zeitlin functions, is the polyhedral cone defined by inequalities (2.1), called the Gelfand–Zeitlin cone.

Let denote the open dense subset of where all inequalities (2.1) are strict (this will be the set in theorem 1.2, cf. conjecture 1.1). The Gelfand–Zeitlin functions are smooth on and define global action coordinates for a completely integrable system: the functions λ⁽ⁿ⁾₁, …, λ⁽ⁿ⁾_n are a complete set of Casimir functions, and the functions λ^(k)_i, 1 ≤ i ≤ k < n, generate

commuting Hamiltonian S¹-actions, whose orbits coincide with the joint level-sets of the Gelfand–Zeitlin functions [7].

Angle coordinates on corresponding to the global action coordinates λ^(k)_i are defined by choosing a Lagrangian section σ of the Gelfand–Zeitlin map and defining ψ^(k)_i(p) = 0 for all p∈Im(σ). The Gelfand–Zeitlin functions are invariant under the transpose map A↦A^T, which is an anti-Poisson involution of . The fixed point set of the transpose map is the set Sym(n) of (real) symmetric n × n matrices and the intersection is a union of Lagrangian submanifolds of that are images of sections of the Gelfand–Zeitlin map. Thus, we may fix global angle coordinates for the Gelfand–Zeitlin system by choosing a connected component of Sym₀(n). In these coordinates, the bivector of the Lie–Poisson bracket on has the form

2.2

In this paper, it will be convenient to introduce the following notation. For all 1 ≤ i ≤ k ≤ n, let

2.3

The functions ℓ^(k)_i satisfy a list of inequalities equivalent to (2.1),

2.4

with the convention that ℓ^(k)₀ = 0 (these inequalities can be visualized as rhombi in a triangular tableau, see [16, figs. 4 and 5]). We denote the interior of the cone defined by (2.4) by . The set is the image under a linear transformation of the interior of the Gelfand–Zeitlin cone. In coordinates ℓ^(k)_i, ψ^(k)_i, the Poisson bivector (2.2) is no longer diagonal; it has a ‘lower-triangular’ form,

2.5

where the coefficients c^(k)_i,j are determined by (2.2) and (2.3).

(b). Poisson–Lie groups and Ginzberg–Weinstein diffeomorphisms

First introduced by Drinfel'd [17] and Semenov-Tian-Shansky [18], a Poisson–Lie group is a Lie group G equipped with a Poisson bivector π such that group multiplication is a Poisson map. The linearization of π at identity of G is a 1-cocycle on with respect to the adjoint representation. This cocycle defines a Lie bracket on and endows the pair with the structure of a Lie bialgebra. The dual Poisson–Lie group of G is the connected, simply connected Poisson–Lie group whose Lie bialgebra is . More details can be found in [19,20].

Let K be a connected compact Lie group. There are two natural Poisson–Lie group structures on K, which in turn define two dual Poisson–Lie groups. We now explain these structures in more detail.

First, let , and fix an Iwasawa decomposition G = KAN, , relative to a choice of maximal torus T⊆K and positive roots. With these choices, and is the direct sum of positive root spaces. Let B( · , · ) be a non-degenerate, K-invariant, bilinear form on , and let be its complexification. The imaginary part of defines a non-degenerate pairing between and which identifies and endows with the Lie algebra structure of . The pair is a Lie bialgebra, which defines a Poisson–Lie group structure on K such that the dual Poisson–Lie group K* is identified with AN.

Second, any Lie group has a trivial Poisson–Lie group structure when equipped with the zero Poisson bivector. The dual Poisson–Lie group of (K, 0) is equipped with the Lie–Poisson structure defined in the previous subsection.

The linearization of the Poisson bivector of K* at the group unit equals the Poisson bivector of . Therefore, K* and are isomorphic as Poisson manifolds in a neighbourhood of their group units by local normal forms for Poisson manifolds [21]. In fact, this isomorphism extends globally.

Theorem 2.1 (Ginzburg–Weinstein Theorem [20]) —

The Poisson manifolds and (K*, π_K*) are Poisson isomorphic.

Such Poisson isomorphisms are called Ginzburg–Weinstein isomorphisms/diffeomorphisms. Note that is abelian, whereas K* is not, so Ginzburg–Weinstein diffeomorphisms cannot be group homomorphisms, and and K* cannot be isomorphic as Poisson–Lie groups.

There are several proofs of the Ginzburg–Weinstein theorem in the literature: the original proof [20] is an existence proof using a cohomology calculation, the proof in [1] gives Ginzburg–Weinstein diffeomorphisms as flows of certain Moser vector fields, the proof in [22] is by integration of a nonlinear PDE of a classical dynamical r-matrix, and the proof in [23] uses the Stokes data of an ODE on a disc with an irregular singular point in the centre.

For the remainder of this section, fix K = U(n) and take the Iwasawa decomposition GL_n = KAN, where A is the set of diagonal matrices with positive real entries and N is the set of upper triangular unipotent matrices. Let denote the set of positive definite n × n Hermitian matrices. The map

2.6

is a diffeomorphism (with inverse given by Gaussian decomposition). It was observed by Flaschka & Ratiu [12] that the functions ln(λ^(k)_i) define a completely integrable system on (equipped with the Poisson structure h_*π_K*). This system was related to the Gelfand–Zeitlin system on by [11] who proved the following theorem.

Theorem 2.2 ([11]) —

There exists a Poisson isomorphism such that

• (i) γ intertwines the Gelfand–Zeitlin functions

  2.7

• (ii) γ intertwines the Gelfand–Zeitlin torus actions on and .

• (iii) For any connected component , .

• (iv) γ is equivariant with respect to the conjugation action of T⊆U(n).

• (v) γ(A + uI) = e^uγ(A).

• (vi) .

Remark 2.3 —

The map γ is uniquely determined by properties (i)–(iii). The map h⁻¹○γ is a Ginzburg–Weinstein diffeomorphism. One should note that [11] use a slightly different h, but the composition h⁻¹○γ remains Poisson. See remark 3.3.

3. Cluster coordinates on dual Poisson–Lie groups

Let K = U(n) and K* = AN as in the previous section. We define coordinates on an open dense subset of AN following [13]: for all 1 ≤ i ≤ k ≤ n, let Δ^(k)_i denote the minor which is the determinant of the solid i × i submatrix formed by intersecting rows n − k + 1 to n − k + i and the last i columns (figure 2).

Figure 2.

The minors Δ^(k)_i.

The minor Δ^(k)_i is a principal minor if and only if i = k, in which case it takes values in , otherwise it takes values in . Let m = n(n − 1)/2; the number of such minors with i < k. Together, the Δ^(k)_i's define a map

whose restriction to the open dense subset where Δ^(k)_i≠0 is a coordinate chart (diffeomorphism). On this subset, the equations

define a t-dependent, polar coordinate chart

where , and is equipped with coordinates (ζ, φ) defined by the equations above. The diffeomorphisms Δ_t define Poisson structures

on .

Let be the interior of the cone defined by inequalities (2.4), replacing ℓ^(k)_i with ζ^(k)_i, and for all δ > 0, define as the subset where

3.1

Theorem 3.1 ([13]) —

For t > 0 and ,

3.2

where C is the number of columns that Δ^(k)_i and Δ^(p)_q have in common, R is the number of rows that Δ^(k)_i and Δ^(p)_q have in common, and ε(x) = x/|x| with ε(0) = 0. Note that δ is constant in the big-O notation O(e^−δt).

This theorem prompts the definition of a Poisson manifold where π_∞ is the constant Poisson structure obtained by taking the t → ∞ limit of π_t, with coefficients given by (3.2). The coordinates ζ⁽ⁿ⁾₁, …, ζ⁽ⁿ⁾_n are a complete set of Casimir functions for π_∞. By [13, theorem 7], there exists a Poisson isomorphism from to , linear with respect to coordinates ℓ, ψ as in §2, of the form

3.3

where B is the integer matrix defining an automorphism of T^m, uniquely determined so that the map is Poisson. Expanding this formula, the isomorphism is of the form

3.4

where ψ^(p)_q is higher than ψ^(k)_i if p < k or p = k and q > i.

Remark 3.2 —

Theorem 3.1 has been generalized in [14]. Given an arbitrary complex semisimple Lie group G and a reduced word for the longest element w₀ of the Weyl group, the coordinate algebra of the double Bruhat cell G^e,w₀ = B∩B − w₀B − carries an upper cluster algebra structure along with a cluster seed [24]. In the coordinates provided by the cluster seed, the Poisson structure on the dual Poisson–Lie group K* () admits a ‘partial tropicalization’: a t-deformation such that the t = ∞ limit is an integrable system , where is the interior of an extended string-cone and π_∞ is a constant Poisson bracket. Setting G = GL_n and taking the standard reduced word for w₀, the cluster seed is given by the minors Δ^(k)_i and one recovers theorem 3.1.

Given a n × n matrix b, and subsets I, J⊆{1, …, n} with |I| = |J|, let Δ_I,J(b) denote the determinant of the submatrix with rows I and columns J. For any 1 ≤ i ≤ n, by the Cauchy–Binet formula,

3.5

where λ_j(bb*) denotes the eigenvalues of bb* (in our notation, these are the Gelfand–Zeitlin functions λ⁽ⁿ⁾_j).

We define a family of Ginzburg–Weinstein diffeomorphisms

3.6

where γ is the Ginzburg–Weinstein diffeomorphism of [11], and h is as defined in §2 (see equation (2.6) and theorem 2.2). For all t > 0,

Denote b_t = gw_t(A). Combining (3.5) and (2.7), for all 1 ≤ i ≤ n,

3.7

Note that if b is upper triangular, the lower right k × k submatrix (bb*)^(k) = b^(k)(b^(k))*, so equation (3.7) remains valid when the eigenvalues λ_j are replaced by the Gelfand–Zeitlin functions λ^(k)_j, and we take I, J⊆{n − k + 1, …, n}.

Remark 3.3 —

If we take instead, then equation (3.7) does not hold for the Gelfand–Zeitlin functions λ^(k)_j, because in general the bottom right k × k submatrix of is not equal to .

Using the generalized Plücker relations, every minor Δ_I,J can be written as a Laurent polynomial in the Δ^(k)_i's (this is a particular example of the Laurent phenomenon in the theory of cluster algebras, see [25]). We can therefore write each |Δ_I,J|² as a Laurent polynomial P_I,J (in variables Δ^(k)_i and )

3.8

Thus, equations (3.7) can be written in the form

3.9

Example 3.4 —

For n = 3, equation (3.9) corresponding to j = 1, k = 3 is

For t large, as λ^(k)_i satisfies the interlacing inequalities, the dominant term on the left side is e^tλ(3)₁. We will see in §4 that if , the dominant term on the right side is e^2tζ(3)₁ = |Δ⁽³⁾₁|².

4. Matrix factorizations and tropical Gelfand–Zeitlin

In this section, we rephrase the results of [16], which uses planar networks, in the language of matrix factorizations.

(a). Matrix factorizations

Let E_ij denote the matrix with (i, j) entry equal to 1 and all other entries 0. For a complex number z, denote . Any word i = (i₁, …, i_k) in the alphabet {1, …, n − 1} defines a map

where N⊆GL_n is the subgroup of upper triangular unipotent matrices. We also define

where A⊆GL_n is the subgroup of diagonal matrices with positive real diagonal entries.

Recall that the Weyl group of GL_n is S_n generated by simple reflections s₁, …, s_n−1. The length l(w) of w∈S_n is the smallest integer k such that w can be written as the product of k simple reflections. A word i = (i₁, …, i_k) is reduced if l(w) = k, where w = s_i1 · ss_ik. Let w₀ be the longest element of S_n with length l(w₀) = m. The standard reduced word for w₀ is

For any reduced word i of w₀, the map, or matrix factorization,

defines a chart, when restricted to . Here, we write (x, z) as shorthand for the tuple (x₁, …, x_n, z₁, …, z_m). When i is the standard reduced word i₀, we write az₀ for simplicity.

Example 4.1 —

For GL₃ and the standard reduced word i₀ = (1, 2, 1), we have:

Remark 4.2 —

More generally, if G is a reductive group with a choice of positive roots, and i = (i₁, …, i_m) is a reduced word for the longest element w₀ of the Weyl group, one can define the maps e_i(z) using a Chevalley basis, and a chart az_i as above. See [26].

Matrix factorization coordinates on AN⊆GL_n can be represented by planar networks; weighted planar graphs, oriented left to right, with n ‘source’ vertices on the left, and n ‘sink’ vertices on the right (see [16] for more details). The matrix factorization az₀ is represented by the standard planar network, Γ, with n horizontal edges connecting the sources to the sinks (both labelled by 1, …, n as in figure 3), and n(n − 1)/2 non-horizontal edges, arranged as in figure 3 for the case n = 3. The non-horizontal edges (which correspond to e_i(z)'s in the matrix factorization) are labelled with the z's and the horizontal edges are labelled at the sources with the x's, as in figure 3. The (i, j) entry of az₀(x, z) equals

where PΓ(i, j) is the set of directed paths in Γ from source i to sink j, e is an edge contained in γ, and w(e) is the weight assigned to e (weights not written on Γ are the multiplicative identity).

Figure 3.

The planar network representing the matrix factorization in example 4.1.

The planar network representation of a matrix factorization gives minors of az_i(x, z) a combinatorial interpretation. Recall from the previous section that Δ_I,J denotes the minor with rows I and columns J, |I| = |J| = i. By the Lindström Lemma [13,27],

where PΓ(I, J) is the set of i-multipaths from I to J; a union of i disjoint directed paths with sources in I and sinks in J. For the minors Δ^(k)_i, there is exactly one such multipath (figure 4).

Figure 4.

Multipaths appearing in the minors of example 4.3. (a) The minor Δ⁽³⁾₂ and (b) the minor Δ_1,2.

Example 4.3 —

Continuing from the previous example,

As illustrated by the example,

Theorem 4.4 ([26, theorem 5.8]) —

For any reduced word i, Δ_I,J(az_i(x, z)) is a polynomial in the coordinates x, z with positive coefficients. Moreover, the minors Δ^(k)_i(az_i(x, z)) are all monomials.

One should note that [26, theorem 5.8] is considerably more general.

(b). Tropical Gelfand–Zeitlin functions

A polynomial is positive if all the coefficients are positive. The tropicalization of a positive polynomial is the piecewise linear function

If the variables x are all real then, substituting x_i = e^tw_i, this equals the limit

If the variables are complex, then we may still define as above, and—on an open dense set—this equals the function obtained by substituting and evaluating the limit as before (the limit does not equal on the subset where leading terms are cancelled due to their complex arguments).

By theorem 4.4, the minors Δ_I,J are positive polynomials in matrix factorization coordinates az_i. Thus, for all 1 ≤ i ≤ k ≤ n, the polynomials

4.1

(which are simply the right sides of equations (3.7)) are positive and can be tropicalized. As the p^(k)_i(x, z) are related, by equation (3.7), to the logarithmic Gelfand–Zeitlin functions on AN, their tropicalizations are called the tropical Gelfand–Zeitlin functions and denoted m^(k)_i. One should note that our definition of m^(k)_i differs from that of m^(k)_i in [15] by a factor of 2.

Theorem 4.5 ([16, theorem 2]) —

For any choice of matrix factorization coordinates az_i, the tropical Gelfand–Zeitlin functions satisfy the rhombus inequalities

4.2

Taken together, the tropical Gelfand–Zeitlin functions define a piecewise linear map, called the tropical Gelfand–Zeitlin map,

Theorem 4.6 ([16, theorem 3]) —

If is defined using the standard reduced word i₀, then (the closure of ).

As is piecewise linear, decomposes into polyhedral chambers where is linear. There is a unique chamber, which we denote by , where the rank of is n + m, and for this chamber, [16]. For all δ > 0, [15] define W^δ to be the set of all such that,

and for any two disjoint subsets α, β⊆EΓ (the edge set of the standard planar network Γ defined in §3),

Let az_i(tw, ϕ) denote az_i(x, z) with the substitutions x_i = e^tw_i, . The condition |w(α) − w(β)| > δ guarantees that there is only one leading term in the polynomials p^(k)_i(az_i(tw, ϕ)), which means that the tropicalization m^(k)_i is equal to the tropical limit described at the beginning of the section. This is the content of the following proposition.

Proposition 4.7 ([16, proposition 2]) —

Fix δ > 0 and let w∈W^δ. For any reduced word i of w₀, there is a constant C such that for all t≥1 and any ϕ∈T^m,

for all 1 ≤ i ≤ k ≤ n.

Let denote the map with coordinates given by

Proposition 4.8 ([16, proposition 5]) —

For every δ > 0, there exists t₀ > 0 such that for all t≥t₀, the following statement holds: for all there exists w ∈ W^δ/2 and ϕ∈T^m such that

We end this section with one last detail from [16] that is crucial in the proof of proposition 5.1. For all , the maximum in the definition of m^(k)_i(w) is the sum corresponding to the monomial Δ^(k)_i(az₀(w, ϕ)) (see also Appendix C of [13]). Thus, for all ,

4.3

5. Convergence of action coordinates

In this section we prove,

Proposition 5.1 —

For all δ > 0 and 1 ≤ i ≤ k ≤ n, there exists a constant C and t₀≥0 such that t≥t₀ implies

5.1

for all .

Proof. —

Let δ > 0, and fix . Denote gw_t(A) = b_t. By definition of ζ^(k)_i and theorem 2.2,

5.2

for all t > 0.

As , by proposition 4.8, there exists t₀ > 0 such that for all t≥t₀, there exists w∈W^δ/2 and ϕ∈T^m such that b_t = az₀(tw, ϕ). As w∈W^δ/2, we can combine equations (5.2), (4.3) and proposition 4.7 to conclude that there exists C≥0 such that

which completes the proof. ▪

6. Convergence of angle coordinates

Recall from §2 that a choice of connected component of determines a choice of angle coordinates ψ for the Gelfand–Zeitlin system. Recall also that (up to a linear transformation), the functions φ are angle coordinates on with respect to π_∞. In this section, we prove there exists a choice of angle coordinates ψ for the Gelfand–Zeitlin system so that,

Proposition 6.1 —

On , for all 1 ≤ i < k ≤ n,

The sum on the right is a linear combination of the coordinates ψ as in equations (3.3) and (3.4) (in particular, ‘higher terms’ refers to the ordering of angle coordinates defined after equation (3.4)).

In §6a, we show that the Hamiltonian vector fields of the Flaschka–Ratiu system on AN converge as t → ∞ to the Hamiltonian vector fields of the action coordinates ζ. In §6b, we prove that for t sufficiently large, the fibres of the Flaschka–Ratiu system intersect the chamber AN + of matrices in AN such that the minors Δ^(k)_i > 0 (one may think of this as the chamber of ‘totally positive’ matrices in AN). These facts are combined in §6c to prove proposition 6.1 and theorem 1.2.

(a). Convergence of Hamiltonian vector fields

For I⊆{1, …, k}, let L_I be the linear function so that (cf. equation (2.3)). Recall also that for I, J⊆{n − k + 1, …, n}, |I| = |J| = j, there exists a Laurent polynomial P_I,J so that

(see equation (3.8)). For all 1 ≤ j ≤ k ≤ n, rearrange equation (3.9) to define functions (for fixed finite t)

6.1

where we have substituted into each P_I,J (see example 3.4).

By theorem 2.2, the functions solve the system of equations

The first step in the proof of proposition 6.1 is to apply the Implicit Function Theorem to this system of equations to get asymptotic control on the partial derivatives of ℓ with respect to ζ, φ.

For both sums on the right side of equation (6.1), there is a single term that dominates exponentially for large t. In the first sum, recall that if , and I≠{1, …, i},

6.2

so the term e^tℓ(k)_j dominates for large t. In the second sum, if , and t > 0 is sufficiently large, then by proposition 4.8, there exists w∈W^δ and ϕ∈T^m such that Δ_t(az₀(tw, ϕ)) = (ζ, φ). Unpacking the proof of [15, Proposition 2], for I, J⊆{n − k + 1, …, n}, |I| = |J| = j, not both equal to {n − k + 1, …, n − k + i},

6.3

so the term |Δ^(k)_j|² = e^2tζ(k)_j dominates the sum for large t. In other words, the j × j minors of the bottom right k × k submatrix are dominated exponentially by the corner minor Δ^(k)_j. The reader may find it useful to work this out for a couple examples, or compare with example 3.4.

In what follows, we order the coordinates

and similarly for ζ and φ.

Lemma 6.2 —

For t > 0 sufficiently large,

where δ_k,p and δ_i,q are Kronecker delta functions.

Note that as the functions f^(k)_j do not involve ζ^(p)'s and φ^(p)'s for p > k, the partial derivatives

Proof. —

For finite t, let with coordinates

as defined in equation (6.1). We will apply the Implicit Function Theorem at solution of f(ℓ, ζ, φ) = 0 with fixed, and

Note that if , then by proposition 5.1, for t sufficiently large.

Order the coordinates ℓ, ζ, φ as above. The matrix of partial derivatives of f with respect to ℓ is the (n + m) × (n + m) block diagonal matrix

where f^(k) = (f^(k)₁, …, f^(k)_k) and ℓ^(k) = (ℓ^(k)₁, …, ℓ^(k)_k). If , then by equation (6.2),

where A^(k) is a k × k matrix whose entries are O(e^−tδ). Thus, for t sufficiently large, D_ℓf is invertible. By the Implicit Function Theorem,

6.4

The function f^(k) does not depend on ζ^(k′)'s for k′ > k. Therefore, the (n + m) × (n + m) matrix

is the block upper-triangular, as is the (n + m) × m matrix (∂f/∂φ). As , by equation (6.3), for each i ≤ k,

where for k = i, B^(k,k) = I + B^(k) is k × k with the entries of matrix B^(k) in O(e^−tδ), and for k > i, B^(k,i) is a k × i matrix with entries in O(e^−tδ). Similarly, for each i ≤ k,

where C^(k,i)_t is a k × (i − 1) matrix with entries in O(t⁻¹e^−tδ) (we absorb a factor of t⁻¹ into C^(k,i)_t to simplify the equation on the next line). Thus,

(note that the ordering of the diagonal entries is the same as for ℓ and f above). Plugging this into equation (6.4), we have

where A is the block diagonal matrix whose diagonal blocks are A^(k). The matrix (I + A)⁻¹ = I + O(e^−tδ), and the result can now be read from the form of this matrix. ▪

In our chart , the Hamiltonian vector field of ℓ^(k)_i(ζ, φ) with respect to π_t is

and the Hamiltonian vector field of ζ^(k)_i with respect to π_∞ is X_ζ^(k)i = {ζ^(k)_i, − }_∞.

Lemma 6.3 —

For all 1 ≤ i ≤ k ≤ n and δ > 0, X_ℓ^(k)i converges uniformly to 2X_ζ^(k)i on .

Proof. —

In matrix notation, the Hamiltonian vector field

Combining theorem 3.1 and lemma 6.2, for and t sufficiently large,

The result follows by proposition 5.1, because e^{t(2ζ(k)_i − ℓ(k)_i)} → 1 as t → ∞. ▪

(b). Totally positive matrices and fibres of the Flaschka–Ratiu system

Let be the set of matrices in AN with real entries. The hypersurfaces defined by equations Δ^(k)_i = 0 divide into chambers. The chamber of ‘totally positive’ matrices is

The restrictions of the functions φ^(k)_i to , defined where Δ^(k)_ℓ≠0, take values in {0, π}. Each chamber of is a joint level set of these functions. By the description above, AN + is the joint level set where every φ^(k)_i = 0.

The set is also divided into chambers by the Flaschka–Ratiu systems (i.e. by the hypersurfaces where singular values ln(λ^(k)_i(bb*)) collide). By theorem 2.2, each of these chambers equals , for a connected component of .

Although the two chamber structures are different, in lemma 6.6 we prove there is a connected component so that for arbitrary δ > 0 and t sufficiently large, the subset is contained in AN + . Here, . As we will see in the next subsection, this implies there is a choice of angle coordinates ψ for the Gelfand–Zeitlin system such that for sufficiently large t, points in the Lagrangian section where ψ = 0 are sent by the scaled Ginzburg–Weinstein map to points in the Lagrangian section where φ = 0.

Lemma 6.4 —

The map gives a 1-1 correspondence between chambers of and coordinates ϕ∈{0, π}^m, i.e. the chambers of equal the images for fixed ϕ∈ {0, π}^m, and these images are distinct.

This lemma implies that the chambers of are connected.

Proof. —

We can prescribe the chamber of containing az₀(x, z) = a(x)z₀(z) by prescribing the z's inductively. We ignore the matrix factor a(x), as it is always positive. As shown in figure 5, label the coordinates z₁, …, z_m by

The minor Δ⁽²⁾₁(z₀(z)) = z_1,2, so we can prescribe its sign by prescribing the sign of z_1,2. Assume we have prescribed the signs of the minors Δ^(p)_q(z₀(z)) by prescribing the coordinates z_q,p, for all 1 ≤ q < p < k. By Lindström lemma, for all 1 ≤ i < k,

Thus, we can prescribe the signs of Δ^(k)_i(z₀(z)) by prescribing the signs of z_i,k. ▪

Figure 5.

Label of variables.

Combining lemma 6.4 and proposition 4.8, for all δ > 0, there exists a t₀ > 0 such that, for all t≥t₀ and , there exists w∈W^δ/2 and ϕ∈T^m, so that

is contained whichever chamber of we please, by choosing ϕ appropriately. Thus, the fibre intersects every chamber of at least once.

Lemma 6.5 —

For all δ > 0, there exists t₀ > 0 such that for all t≥t₀ and , the fibre intersects every chamber of exactly once.

Proof. —

Fix δ > 0 and let t₀ as in proposition 4.8. The fibre L⁻¹(ℓ) intersects Sym(n) at exactly 2^m points (this follows directly from linear algebra). By theorem 2.2, , and if and only if x∈Sym(n). Thus, the fibres of intersect at exactly 2^m points. As there are 2^m chambers in , and intersects every chamber of at least once, this completes the proof. ▪

Next, we show that there is a unique connected component whose elements are sent to AN + by gw_t for t sufficiently large. For a connected component , let .

Lemma 6.6 —

There is a unique connected component such that for all δ > 0, there exists a t₀≥0 such that for all t≥t₀ and ,

Proof. —

Fix δ > 0 arbitrary. By lemma 6.5, there exists t₀≥0 such that for all t≥t₀ and , intersects every chamber of exactly once.

Consider the set

where (recall L is the Gelfand–Zeitlin map with coordinates ℓ^(k)_i). By lemma 6.5, the image of this set under L equals .

If there exists disjoint open subsets A, B⊆Sym^δ(n), so that

then

As the restriction of L to is open as a map to , L(A) and L(B) are open. As for all , intersects AN₊ exactly once, the sets L(A) and L(B) are disjoint. This implies that is not connected, which is a contradiction (it is convex). Thus, the set gw⁻¹_t0(AN + )∩Sym^δ(n) is connected, and must be contained in a connected component .

The result follows for all t≥t₀, since and for all t≥1. Explicitly, given t≥t₀ and , we have by definition of gw that

 ▪

(c). Proof of proposition 6.1 and theorem 1.2

Proof of proposition 6.1. —

Fix an arbitrary regular fibre of the Gelfand–Zeitlin system, L⁻¹(ℓ₀). As , for some δ > 0. We will show the functions φ^(k)_i○Δ_t○gw_t converge on L⁻¹(ℓ₀) by showing that they converge at a point in L⁻¹(ℓ₀) and their derivatives converge uniformly on L⁻¹(ℓ₀).

Let be the unique connected component described in lemma 6.6. Choose angle coordinates ψ for the Gelfand–Zeitlin system so that . Let x be the unique point in that has coordinates ψ(x) = 0. By lemma 6.6, for all t sufficiently large, gw_t(x)∈AN + . Thus, for all t sufficiently large and 1 ≤ q < p ≤ n,

which equals the value of the linear combination ψ^(p−1)_q + ‘higher terms’ at x for our choice of angle coordinates.

Second, for all A∈L⁻¹(ℓ₀), 1 ≤ i ≤ k, and for t sufficiently large,

6.5

By lemma 6.3, this converges uniformly on any to the constant function

6.6

As Δ_t○gw_t(A) is contained in for t sufficiently large, this completes the proof. ▪

Proof of theorem 1.2. —

As observed in the proof of [13, theorem 7], there is a unique Poisson isomorphism from equipped with to equipped with π_∞ such that

and

Combining propositions 5.1 and 6.1, we see that in coordinates ζ, φ on the map (1.1) converges on to this Poisson isomorphism. This is the Gelfand–Zeitlin system, up to a linear change of coordinates. ▪

7. Conclusion

Recall from §4 that the tropicalization of a positive polynomial p(x), in complex variables, is equal, on an open dense set, to the ‘tropical’ limit

where we have substituted . The scaling limit of Ginzburg–Weinstein diffeomor- phisms studied in this paper may be viewed as a non-abelian ‘tropical limit’: Ginzburg–Weinstein diffeomorphisms can be written as the composition

where , and f(A) = Ad^*_Ψ(A)A, , is the flow of a Moser vector field on [1,11]. Using this factorization, we can rewrite the limit of proposition 5.1 in the more suggestive form,

where p = p^(k)_i is a positive polynomial in matrix factorization coordinates, and we have suppressed the diffeomorphism P≅AN.

We hope that a more general theory of non-abelian tropical limits including Ginzburg–Weinstein maps for compact Lie groups other than U(n) will emerge.

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Author's contributions

All authors contributed equally to the main results. J.L. and Y.L. drafted the manuscript. All authors read and approved the manuscript.

Competing interests

The authors declare that they have no competing interests.

Funding

Our work was supported in part by the project MODFLAT of the European Research Council (ERC), by the grant nos. 178794 and 178828 of the Swiss National Science Foundation (SNSF) and by the NCCR SwissMAP of the SNSF.