With Wronskian through the looking glass

In the work of Mukhin and Varchenko from 2002 there was introduced a Wronskian map from the variety of full flags in a finite dimensional vector space into a product of projective spaces. We establish a precise relationship between this map and the Pl\"ucker map. This allows us to recover the result of Varchenko and Wright saying that the polynomials appearing in the image of the Wronsky map are the initial values of the tau-functions for the Kadomtsev-Petviashvili hierarchy.

On the other hand, in [MV] there was introduced a map W : X −→ (P N ) n−1 (N being big enough) which we call the Wronskian map since its definition uses a lot of Wronskians. This map has been studied in [SV]. We will see below that W lands in a subspace n−1 i=1 P (i(n−i)−1 ⊂ (P N ) n−1 , so we will consider it as a map W = (W 1 , . . . , W n−1 ) : X −→ P ′ := n−1 i=1 P (i(n−i)−1 . (0. 2) The present note, which may be regarded as a postscript to [SV], contains some elementary remarks on the relationship between Pℓ and W.
We define for each 1 ≤ i ≤ n − 1 a linear contraction map For g ∈ G letḡ ∈ X denote its image in X; let W i (g) = (a 0 (g) : . . . : a i(n−i) (g)), and consider a polynomial As a corollary of Theorem 2.3 we deduce that the polynomials y i (g) are nothing else but the initial values of the tau-functions for the KP hierarchy, see Theorems 3.4.2 and 3.5.2; this assertion is essentially [VW], Lemma 5.7.
As another remark we reinterpret in §4 the W 5 identity instrumental in [SV] as a particular case of the classical Desnanot -Jacobi formula, and explain its relation to Wronskian mutations studied in [MV] and [SV].
We are grateful to A.Kuznetsov for a useful discussion. §1. Wronsky map 1.1. We fix a base commutative ring k ⊃ Q.

Let
f = (f 1 (t), . . . , f n (t)) be a sequence of rational functions f i (t) ∈ k(t). Its Wronskian matrix is by definition an n × n matrix W(f) = (f In general we identify the space of polynomials of degree ≤ m with the space of k-points of an affine space: Note that if n = m then It follows that if e ij (a), i > j, is a lower triangular elementary matrix then It follows that for any A ∈ N − (k) (a lower triangular with 1's on the diagonal) On the other hand, if In other words, if A ∈ B − (n, k) (the lower Borel), which is of course is seen immediately.

Degrees and Bruhat decomposition
Suppose that n = m. We will denote by G = GL n , B − ⊂ G the lower triangular Borel, N − ⊂ B − etc.
It turns out that d(g) can take only n! possible values situated in vertices of a permutohedron.
Namely, consider the Bruhat decomposition Identify N n−1 with the root lattice Q of G s := SL n using the standard base {α 1 , . . . , α n−1 } ⊂ Q of simple roots.
Here ∆ ij (g) denotes the 2 × 2 minor of g picking the first two rows and i-th and j-th columns.
One checks directly that This means that g ∈ B − iff a 12 = a 13 = ∆ 23 (g) = 0. Similarly (the big cell). This means that These formulas may be understood as a criterion of recognizing Bruhat cells in GL 3 , cf. [FZ].
Therefore the Wronskian map induces maps for each w ∈ W .

Induced map on the flag space
The invariance (1.2.2) implies that W induces a map from the base affine space More explicitly: we can assign to an arbitrary matrix g = (b ij ) ∈ G a flag in V = k n It is clear that F (g) = F (ng) for n ∈ B − , and the map where Fℓ(V ) is the space of full flags in V , induces an isomorphism On the other hand consider the restriction of F to the upper triangular group We may also consider the composition (1.4.5) We will see below (cf. 2.8) that this map is an embedding.  Let P i ⊂ G = GL n denote the stabilizer of the coordinate subspace A i ⊂ A n , so that the Grassmanian of i-planes in A n . The lemma of Bruhat gives rise to an isomorphism This set may also be interpreted as "the set of F 1 -points" with n ≤ m. For any j ∈ [n] M ≤j will denote the truncated matrix We suppose that rank(M) = n.
For any j ∈ [n] consider the set of j × j minors of M or the same set up to a multiplication by a scalar We denoteP and Pℓ ( We will compare it with the i-th component of the Wronskian map To formulate the result we will use the Schubert decomposition from 2.1 (a).
Let p = ℓ(w) denote the length of a minimal decomposition w = s j 1 . . . s jp into a product of Coxeter generators s j = (j, j + 1).
Sometimes it is convenient to depict elements of C i n as sequences (2.2.1) In this notation This is a particular case of a more general statement, see below 2.5.2.
For each j ≥ 0 consider the subset The following statement is the main result of the present note.

From
Pℓ to W: a contraction.
where the numbers m(I) ∈ Z >0 are defined below see 2.5.
In other words, the map l induces a contraction map We will also use below the reciprocal polynomials Proof of 2.3 is given below in 2.7 after some preparation.
(iii) Let again n = 4. Then (see [SV], (5.11)) We call i = i p admissible if either p = k and i p < n or i p+1 > i p + 1. We denote by I o ⊂ I the subset of admissible elements.
The reader should compare this definition with operators defining a representation of the nil-Temperley-Lieb algebra from [BFZ], (2.4.6), cf. also [BM] and references therein.

"Balls in boxes" picture
Recall the representation (2.2.1) of elements of C k n : where we imagine the 1's as k "balls" sitting in n "boxes". An operation ∆ i means moving the ball in i-th box to the right, which is possible if the (i + 1)-th box is free.
Each I ∈ C k n may be written as for some j 1 , . . . , j p . This is clear from the balls in boxes picture.
Here l(I) is from 2.2.1.
Proof. Clear from BB picture.
To put it differently, define a graph Γ k n whose set of vertices is C k n , the edges having the form I −→ ∆ i I (or otherwise define an obvious partial order on C i n ). Then m(I) is the set of paths in Γ k n going from the minimal element [k] to I. 2.5.3. Symmetry. This graph can be turned upside down.
Clear from the "balls in boxes" description.
2.6. Generalized Wronskians and the derivative. Let be a sequence of functions. We can assign to it a ∞ × ∞ Wronskian matrix . . , i k and columns 1, 2, . . . , k, and let

By definition
For I, J ⊂ [n] let M IJ (g) denote the submatrix of g lying on the ntersection of the lines (columns) with numbers i ∈ I (j ∈ J), so that We see that the constant term where M t denotes the transposed matrix.
To compute the other coefficients we use Lemma 2.6.1 and Corollary 2.6.2. So and more generally y (p) which implies the formula.
2.8. Triangular theorem. Consider an upper triangular unipotent matrix g ∈ N ⊂ GL n (k). We claim that g may be reconstructed uniquely from the coefficients of polynomials y 1 (g), . . . , y n−1 (g).
More precisely, to get the first i rows of g we need only a truncated part of the first i polynomials (y 1 (g) = y 1 (g) ≤n−1 , y 2 (g) ≤n−2 , . . . , y i (g) ≤i ) This is the contents of [SV], Thm 5.3. We explain how it follows from our Thm 2.3.
To illustrate what is going on consider an example n = 5. Let We have y 1 (g) = b 1 (g), so we get the first row of g, i.e. the elements a i , from y 1 (g).
Triangular structure on the map W N We can express the above as follows. Let B := W(N), so that Obviously N ∼ = k n(n−1)/2 ; we define n(n − 1)/2 coordinates in N as the elements of a matrix g ∈ N in the lexicographic order, i.e. n − 1 elements from the first row (from left to right), n − 2 elements from the second row, etc.

Let
W(g) = (y 1 (g), . . . , y n−1 (g)); we define the coordinates of a vector W(g) similarly, by taking n − 1 coefficients of y 1 (g), then the first n − 2 coefficients of y 2 (g), etc. The exact value of n(I) will be given below, see 3.3.1.

Electrons and holes
Let C ∞/2 denote the set of subsets S = {a 0 , a 1 , . . . , } ⊂ Z such that both sets S(ν) \ N, N \ S(ν) are finite, a i = i for i sufficiently large. Its elements enumerate the cells of the semi-infinite Grassmanian, cf. [SW].
We can imagine such S as the set of boxes numbered by i ∈ Z, with balls put to the boxes with numbers a j .
We define the virtual dimesnsion by  note that a n = n implies a m = m for all m ≥ n.
Note that we have a bijection

2.2)
obvious from the "balls in boxes" picture.
Namely, we have inside C 0 ∞/2,n the minimal state S min = (a i ) with a i = 1 for − n ≤ i ≤ −1 and for i ≥ n from which one gets all other states in C 0 ∞/2,n by moving the balls to the right, until we reach the maximal state we will denote by ν(I) the corresponding partition.

Transposed cells.
For I ∈ C i n let I t = I ′ ∈ C i n denote the "opposite or transposed cell which in "balls and boxes" picture it is obtained by reading I from right to left.
The corresponding partition λ(I t ) = λ(I) t has the transposed Young diagram.

Initial Schur functions and the Wronskian. Let us introduce new coordinates
cf. [SW] (8.4).

For example
etc.
Let us consider the Schur functions s ν as functions of t i . The first coordinate x = t 1 is called the space variable, whereas t i , i ≥ 2, are "the times".
We will be interested in the "initial" Schur functions, the values of s ν (t) for t 2 = t 3 = . . . = 0.
On the other hand suppose that ν = ν(I) for some Proof. (i) is [SW], proof of Prop. 8.6.
(ii) is a consequence of a more general Claim 3.5.0 below. Then:

Example. Let
(i) 3.4. Polynomials y n (g)(x) and initial tau-functions: the middle case.
Let n ≥ 1. For let ν(I) denote the corresponding partition.
For a matrix g ∈ GL 2n we define its tau-function τ (g) which will be a function of variables t 1 , t 2 , . . . , by cf. [SW], Prop. 8.3.

Differential equation
Suppose for simplicity that a = a 12 = 1, introduce the notation x = t 1 for the space variable, so τ (g) = 1 + ax. Let It satisfies a differental equation which is the stationary KdV.
with Schur functions: Thus for g ∈ GL 4 , τ 2 (g) has 6 summands: where ∆ ij (g) indicates the minor with i-th and j-th columns.
Here we use the reciprocal polynomialsỹ i (g) defined in (2.3.1).
Proof. We use the definition (3.4.1): The assertion 3.4.2 follows from this.
3.5. Case of an arbitrary virtual dimension. Let i ≤ n. Consider an embedding where d(n, i) is chosen in such a way that ν(I max ) = () (this defines d(n, i) uniquely).
Here ν(I) denotes the partition corresponding to I ∈ C i n under the composition where the last isomorphism is a shift (a j ) → (a j−i ), and identifying C 0 ∞/2 with the set of partitions.
More precisely, for any d a sequence S = (a 0 , a 1 , . . .) belongs to C d ∞/2 iff a i = i − d for i >> 0.
To such S there corresponds a partition Let I ∈ C i n . Then s ν(I) (h) = ∆ I t (T ).
where I t ∈ C i n is the transposed cell (see 3.2.4). Note that for any i and any I ∈ C i n the function s ν(I) (h) depends exactly on h 1 , . . . h n−1 . Afterwards we can apply the same construction as above: to g ∈ GL 2n+i we assign a tau-function
So for a matrix g = (a ij ) ∈ GL n its first tau-function as it should be, the contraction map Plücker −→ Wronsky being the identity. Whence For a matrix g = (a ij ) ∈ GL n its (n − 1)-th tau-function 3.6. Wronskians as τ -functions. One can express the above as follows. Consider a Tate vector space of Laurent power series It is equipped with two subspaces, H + = k [[z]] and Let Gr = Gr ∞/2 ∞ denote the Grassmanian of subspaces L ⊂ H of the form [SW], §8.
In other words, such L should admit a topological base of the form 3.6.1. For example, in 3.5.3 above we see a description of embeddings P n−1 = Gr 1 n ֒→ Gr, (P n−1 ) ∨ = Gr n−1 n ֒→ Gr To each L ∈ Gr there corresponds a tau-function cf. [SW], Proposition 8.3.
Given a sequence of polynomials This statement is a reformulation of 3.4 and 3.5.
3.7. Full flags and MKP. For n ∈ Z ≥2 define a semi-infinite flag space Fℓ ∞/2 n whose elements are sequences of subspaces This is a subspace of the semi-infinite flag space considered in [KP], §8, whose elements parametrize the rational solutions of the modified Kadomtsev-Petviashvili hierarchy.
coincides with the initial value τ F (g) (x, 0 . . .) of the τ -function τ F (g) of the MKP hierarchy corresponding to the semi-infinite flag F (g).
This is an immediate consequence of the Grassmanian case.
Denote by A 1n,1n the (n − 2) × (n − 2) submatrix obtained from A by deleting the first and the n-th row and the first and the n-th column, etc.
Then det A det A 1n,1n = det A 1,1 det A n,n − det A 1,n det A n,1 (4.2.1) 4.3. Let us rewrite (4.1.1) in the form we see that it resembles (4.2.1).
And indeed, if we apply (4.2.1) to the Wronskian matrix W(A ∪ B), we get (4.1.1). This remark appears in [C], the beginning of Section 3.
We leave the details to the reader.

Example: Lusztig vs Wronskian mutations.
Recall the situation 1.2, where we suppose that m = n.
The main result of [SV] describes the compatibility of the map W with multiplication by an upper triangular matrix e i,i+1 (c) = I n + ce ij , e ij = (δ pi δ qj ) p,q ∈ GL n (k) The following example is taken from [SV], proof of Thm 4.4. is called a Wronskian mutation equation. It is regarded in [SV] as a differential equation of the first order on an unknown functionỹ 2 (c, M) = y 2 (e 23 (c)M); together with some initial conditions it determines the polynomialỹ 2 (c, M) uniquely from given y i (M), 1 ≤ i ≤ 3.
This equation is a modification of a similar differential equation from [MV] whose solution is a functionỹ