Birational maps to Grassmannians, representations and poset polytopes

We study the closure of the graph of the birational map from a projective space to a Grassmannian. We provide explicit description of the graph closure and compute the fibers of the natural projection to the Grassmannian. We construct embeddings of the graph closure to the projectivizations of certain cyclic representations of a degenerate special linear Lie algebra and study algebraic and combinatorial properties of these representations. In particular, we describe monomial bases, generalizing the FFLV bases. The proof relies on combinatorial properties of a new family of poset polytopes, which are of independent interest. As a consequence we obtain flat toric degenerations of the graph closure studied by Borovik, Sturmfels and Sverrisd\'ottir.


Introduction
Let Gr(d, n) be the Grassmannian of d-dimensional subspaces in an ndimensional vector space.The Grassmann variety admits a cellular decomposition into Bruhat cells with a unique open cell A N , where N = d(n − d) is the dimension of Gr(d, n).Hence one gets the birational exponential map ı : P N → Gr (d, n).Our main object of study is the closure of the graph of this map, i.e. the variety G(d, n) ⊂ P N × Gr(d, n), which is the closure of the set (x, ı(x)) for x ∈ A N .Some results about these varieties were obtained in [BSS, FSS] (see also [KP] for the symplectic case).
By definition, there is a surjective map G(d, n) → Gr(d, n).Our first task is to describe the fibers of this map.To this end, let us denote by L the ndimensional vector space; any U ∈ Gr(d, n) is a d-dimensional subspace of L. We fix a decomposition L = L − ⊕L + , where dim L − = d and dim L + = n−d.Consider the stratification Gr(d, n) = min(d,n−d) k=0 X k , where X k consists of subspaces U ⊂ L such that dim(U ∩ L + ) = k and X 0 is the image of the map ı : A N → Gr(d, n).We prove the following theorem.
Theorem 0.1.The map ϕ is one-to-one over X 0 ⊔ X 1 .For k ≥ 2 the map ϕ over X k is a fiber bundle with a fiber isomorphic to the projective space P k 2 −1 .
In order to study the varieties G(d, n) we consider the action of a certain group SL (d) n on G(d, n).This group is a degeneration of the classical group SL n depending on the parameter d, it contains a normal abelian subgroup N d (isomorphic to an abelian unipotent radical of SL n ) and the maximal standard parabolic subgroup P d of SL n (corresponding to the d-th simple root).More precisely, SL  In particular, G(d, n) are the G N a varieties (see [Ar,AZ,F1,HT]).We note that the modules L m,M are defined for all pairs m, M ∈ Z ≥0 , and for m = 0 or M = 0 the map G(d, n) → P(L m,M ) still exists, but is not an embedding.The spaces L m,M are responsible for the description of the homogeneous coordinate rings of G(d, n) with respect to the generalized Plücker embeddings and for the spaces of sections of natural line bundles, so we are interested in algebraic and combinatorial properties of L m,M .More precisely, we want to describe L m,M as sl (d) n modules (via explicit relations), find monomial bases and compute the dimensions.
To this end, we generalize the construction of the FFLV bases (see [FFL1]).Namely, we consider the set of roots P of a d and for α ∈ P we denote by f α ∈ a d the corresponding root vector.The set P consists of roots α i,j with 1 ≤ i ≤ d ≤ j ≤ n − 1 and has a natural structure of a poset.In particular, the Dyck paths from [FFL1] are chains in P .We define the set S(m, M ) ⊂ Z P ≥0 consisting of collections (s α ) α∈P subject to the conditions: (0.1) where w(P ′ ) is the width of the subposet P ′ (the minimal number of chains needed to cover P ′ ).We prove the following theorem.
Theorem 0.3.The elements α∈P f sα α , (s α ) α ∈ S(m, M ) being applied to the cyclic vector form a basis of L m,M .
Theorem 0.4.Let P be a finite poset.Then X(m, M ) are lattice polytopes such that In particular, the polytopes X(m, M ) have the integer decomposition property (see [HOT, T]).As an application of Theorem 0.4 we obtain that the polytopes X(m, M ) define flat toric degenerations of G(d, n) (see [An,AB,FaFL1,O]).Our approach is in some sense dual to the one used in [BSS], where the Plücker type relations for G(d, n) are utilized.We note that in [BSS] the authors consider the Cayley sums, which are closely related to the Minkowski sums (see e.g.[T]).
Let us close with the following remark.There are two natural directions we plan to address elsewhere.First, one can replace the Grassmannians with arbitrary flag varieties of type A or their PBW degenerate versions (see [ABS,FaFL2,F1,F2,FeFi] for the corresponding results on the PBW degenerations).Second, it is interesting to consider symplectic and orthogonal cases where the PBW structures are also available (see [BK,FFLi,FFL2,G,M1]).
The paper is organized as follows.In Section 1 we introduce the polytopes X(m, M ) and study their properties.The proof of the main theorem is given in Appendix A. In Section 2 we recall the construction of the FFLV bases in the irreducible highest weight representations.We also introduce the modules L m,M and study their algebraic properties using the polytopes X(m, M ).In Section 3 we introduce the graph closures G(d, n) and describe the fibers of the natural projection map G(d, n) → Gr(d, n).We use the results from the combinatorial and algebraic sections to describe the geometric properties of G(d, n).

Combinatorics
In this section we attach a family of polytopes X(m, M ), m, M ∈ Z ≥0 to a finite poset P and show that these polytopes satisfy Minkowsky property with respect to the parameters.The polytopes X(m, M ) play a crucial role in the next sections.
Let P be a finite poset.For a subposet P ′ ⊂ P we denote by w(P ′ ) the width of P ′ , which can be defined as a minimal number of chains needed to cover P ′ or, equivalently, as the maximal length of an antichain in P ′ (Dilworth's theorem [D]).For two nonegative integers m and M we define a polytope X(m, M ) ⊂ Z P ≥0 consisting of points (x α ) α∈P subject to the conditions (1.1) We denote by S(m, M ) the set of integer points in X(m, M ).
Example 1.1.Let m = 0. Then X(0, M ) is a scaled simplex of P (the sum of all coordinates does not exceed M ).
Example 1.2.Let M = 0. Then X(m, 0) is a scaled chain polytope of P (see [Stan]).In fact, the defining inequalities are α∈P ′ x α ≤ m • w(P ′ ) for all P ′ ⊂ P , which are implied by the subset of inequalities with w(P ′ ) = 1 (i.e. for P ′ being a single chain).Therefore, X(m, 0) consists of elements (x α ) α∈P such that the sum over any chain does not exceed m.Hence X(m, 0) = m∆ chain (P ), where ∆ chain (P ) is the Stanley chain polytope.In particular, the points of S(1, 0) are the indicator functions of antichains in P and S(m, 0) are all possible sums of m indicator functions.
Proof.We note that S(m Hence it suffices to show that S(m, M ) + S(0, 1) = S(m, M + 1).This is shown in Theorem A.1.
Corollary 1.4.The polytopes X(m, M ) are normal lattice polytopes.For any nonnegative m 1 , m 2 , M 1 , M 2 one has Proof.Assume that p is a rational vertex of X(m, M ).Then there exists r > 0 such that rp has integer coordinates and hence rp ∈ S(rm, rM ).Now Theorem 1.3 implies that rp is equal to a sum of r points in S(m, M ): Since rp is a vertex of X(rm, rM ), we conclude that p i = p and hence p ∈ S(m, M ).So X(m, M ) are lattice polytopes.
Let us derive a corollary from Theorem 1.3.
Corollary 1.5.Let P be a finite poset such that for some m, M ∈ Z ≥0 one has Then there exists a decomposition of P into the union of disjoint sets where A i are antichains and |B| ≤ M .
Proof.To show that Theorem 1.3 implies Corollary 1.5 it suffices to take a point s ∈ Z P ≥0 such that s α = 1 for all α ∈ P .By definition, s ∈ S(m, M ) and hence there exist s 1 , . . ., s m ∈ S(1, 0) such that s − m i=1 s i ∈ S(0, M ).Now A i is the support of s i and B is the support of s− m i=1 s i ∈ S(0, M ).

Algebra
In this section we introduce and study main algebraic objects we are interested in.In particular, the polytopes from the previous section are used to construct monomial bases in certain cyclic representations of abelian algebras.
2.1.The PBW setup.We fix nonnegative integers n and d such that The same basis of a d can be described in terms of root vectors.Namely, let Let ω i , i ∈ [n − 1] be the fundamental weights.For a dominant integral weight λ = m i ω i we denote by L λ the corresponding irreducible highest weight sl n module with a highest weight vector ℓ λ ∈ L λ .In this paper we deal with λ = mω d .In particular, L ω d ≃ Λ d (L), where L = L ω 1 is the n-dimensional vector representation.We fix a basis ℓ 1 , . . ., ℓ n of L; for We note that the vector For an arbitrary m > 0 the irreducible module L mω d sits inside the tensor power L ⊗m ω d as a Cartan component.More precisely, Since a d is abelian, each representation L mω d is naturally identified with a quotient of the polynomial ring C[f i,j ], i ≤ d ≤ j by a certain ideal.We note that this ideal is graded with respect to the total degree of polynomials in C[f i,j ] (the degree of each variable f i,j is one).Hence we get the induced PBW grading on L mω d : In other words, the degree of a wedge monomial ℓ I is equal to the number of elements of I outside [d] (this the PBW degree in the terminology of [FFL1] or the level in the terminology of [FO, FSS]).
In [FFL1] (see also [Vin]) the authors construct monomial bases for the finite-dimensional irreducible highest weight representations of sl n .We recall the construction below (for highest weights mω d ).A Dyck path p = (p 1 , . . ., p n−1 ) is a subset of the set α i,j , i ≤ d ≤ j satisfying the following conditions: • Remark 2.2.Recall that the roots of a d are naturally identified with matrix units: f i,j = E j+1,i .Via this identification, a Dyck path starts at E d+1,1 , ends at E n,d and at each step goes either one cell down or one cell right.
Consider the following partial order on R(d) : .
Example 2.4.Let m = 1.Then the points in S ω d are in one-to-one correspondence with antichains in R(d).In fact, the antichains in R(d) are of the form Let p d ⊂ sl n be the standard maximal parabolic subalgebra corresponding to the radical a d .More precisely, p d is spanned by the matrix units E i,j such that either i ≤ d or j > d (with an extra traceless condition).One has sl n = a d ⊕ p d .Since p d is a subalgebra in sl n , any representation of sl n is naturally a module over the parabolic subalgebra.One easily sees that and hence we obtain an induced p d action on the module Remark 2.5.We note that grL mω d is naturally isomorphic to L mω d as a vector space and as a d modules.We use gr to stress that the p d action is different.In particular, grL mω d is a direct sum of p d submodules L mω d (r), while the summands L mω d (r) of L mω d are not p d invariant.
Lemma 2.6.The induced actions of a d and p d on grL mω d glue into the action of the degenerate algebra sl Proof.Immediate from the definitions (see also [FFL1,F3,PY]).
2.2.The extension.Let us consider the following graded module V of the abelian Lie algebra a d .As a vector space V = V (0) ⊕ V (1), where V (0) is one-dimensional space spanned by vector v and V (1) ≃ a d as a vector space.The abelian Lie algebra a d acts trivially on V (1) and sends V (0) to V (1) via the identification V (0) ⊗ a d ≃ V (1).Then V is generated from v by the action of a d and V is isomorphic to the quotient of the polynomial algebra in variables f i,j , 1 ≤ i ≤ d ≤ j ≤ n − 1 by the ideal generated by all monomials of degree two.
Similarly, we define the a d modules V M , M ≥ 1 as the quotient of C[f i,j ] by the ideal generated by all degree M +1 monomials (in particular, V ≃ V 1 ).Then V M ≃ V ⊙M 1 as a d modules and Lemma 2.7.The a d action on V M extends to the sl Proof.We note that a d acts on V M simply by multiplication.Now the claim follows from the definition of the action of p d .
We are now ready to define the main algebraic object of this paper.2.3.Bases.We consider the real vector space R d(n−d) whose coordinates are labeled by the roots α i,j , 1 ≤ i ≤ d ≤ j < n of a d , or simply by the corresponding pairs i, j.
is the set of points (s i,j ) with nonnegative coordinates satisfying the following inequalities for all r ≥ 1: (2.2) α∈p 1 ∪...∪pr s α ≤ rm + M for all tuples of Dyck paths p 1 , . . ., p r .
Proof.We note that if r = min(d, n − d), then there exist r Dyck paths p 1 , . . ., p r such that r i=1 p i cover the whole set of roots α i,j , i ≤ j ≤ d.In fact, without loss of generality we assume that d ≤ n − d (i.e.r = d).Then we take The inequality corresponding to the paths p 1 , . . ., p r reads as i≤d≤j s i,j ≤ md + M , which implies all inequalities (2.2) for r > d.
Our goal is to show that the following properties hold: Lemma 2.16.The three properties above hold true for M, M ′ = 0.
Proof.The polytope X(m, 0) is the FFLV polytope for the weight mω d .Hence the first two properties are direct corollaries from [FFL1].To prove the third property we note that if M = 0 then all the relations are implied by the r = 1 relations.In fact, all f i,j commute and b 1 + • • • + b r > rm implies that b k > m for some k = 1, . . ., r. However it was shown in [FFL1] that the defining relations of grL mω d are f m+1 i,j ℓ mω d = 0 (in fact, it suffices to take f m+1 1,n−1 ℓ mω d = 0).Lemma 2.17.The relations The claim of the Lemma is implied by two observations: In fact, when applying f b 1 i 1 ,j 1 . . .f br ir,jr to ℓ m,M one distributes the factors of the monomial among the factors of the tensor product ℓ ⊗m In this case b i > m for some i.However, f 2 i,j ℓ ω d = 0 and hence f m+1 i,j ℓ ⊗m ω d = 0. Proposition 2.18.The vectors f s ℓ m,M , s ∈ S(m, M ) span L m,M .
Proof.Thanks to Lemma 2.17 it is enough to show that any vector f t ℓ m,M , t / ∈ S(m, M ) can be rewritten as a linear combination of vectors f s ℓ m,M , s ∈ S(m, M ) using relations from Lemma 2.17 (as we mentioned in Remark 2.15 we use all the relations (2.3) and their p d consequences).
Recall the idea of the proof of a similar statement from [FFL1], which is the M = 0, r = 1 case.Let us (totally) order the root vectors as follows: (2.4) In other words, f i,j ≻ f i ′ ,j ′ if and only if j > j ′ or (j = j ′ and i > i ′ ); note that this total order refines the partial order from Remark 2.4.In what follows we consider the associated homogeneous (with respect to total degree -the sum of all exponents) lexicographic order on the set of monomials f t .It was shown in [FFL1] that if t / ∈ S(m, 0), then f t ℓ m,0 can be rewritten as a linear combination of smaller monomials (of the same total degree).Since there are obviously only finitely many monomials smaller than a given one, we obtain that f s ℓ m,0 , s ∈ S(m, 0) span the space L m,0 .We generalize this argument to the case of general M .
We prove that if t does not satisfy some inequality from (2.2), then f t can be rewritten as a linear combination of smaller monomials (with respect to the order induced by (2.4)).Assume we are given an r-tuple of Dyck paths p 1 , . . ., p r .We assume that the union of these paths is of width r (i.e. the union can not be covered by a smaller number of paths).We represent p 1 ∪ • • • ∪ p r as a union of r pairwise nonintersecting sets p 1 , . . ., p r with the following properties: • each p k is a dense chain (i.e. the are no elements between two neighbors of p k , hence by Lemma 2.17 one has: f b 1 i 1 ,j 1 . . .f br ir,jr ℓ m,M = 0. Now our goal is to show that U(sl i 1 ,j 1 . . .f br ir,jr ℓ m,M contains a vector which is equal to f t ℓ m,M plus some linear combination of smaller monomials applied to ℓ m,M .
We follow the strategy of [FFL1].More precisely, we consider the derivations ∂ α of the polynomial ring in variables f β defined by is a root and zero otherwise.We denote by ∂ i,j the operators ∂ α i,j .We start with f br ir,jr and the Dyck path p r .Applying certain linear combination of the products of operators ∂ α to f br ir,jr we obtain α∈p r f tα α plus linear combination of smaller monomials.An important thing is that we only use the operators of the form ∂ •,jr and ∂ ir,• .Therefore, since i 1 < • • • < i r and j 1 > • • • > j r , the operators ∂ α we use at this stage do not affect f b k i k ,j k for k < r.As a result we obtain that can be expressed as a linear combination of smaller monomials applied to ℓ m,M .
We then proceed with f b r−1 i r−1 ,j r−1 and the Dyck path p r−1 .As in the previous case, we apply the derivation operators to f b r−1 i r−1 ,j r−1 and obtain α∈p r−1 f tα α modulo a linear combination of smaller terms.The derivation we use at this stage do not affect f ba ia,ja for a < r − 1 and do not affect the result of application of derivation operators to f br ir,jr from the previous stage.As a result at this point we obtain that can be expressed as a linear combination of smaller monomials applied to ℓ m,M .
We proceed in the same way with f b r−2 i r−2 ,j r−2 , etc. and as a result obtain that can be expressed as a linear combination of smaller monomials applied to ℓ m,M .Hence the same property holds true for the whole monomial f t ℓ m,M , as desired.
Theorem 2.19.Vectors f s ℓ m,M , s ∈ S(m, M ) form a basis of L m,M and relations (2.3) are defining for the cyclic sl Proof.We use the results of [FFL3].The bases f s ℓ 1,0 , s ∈ S(1, 0) and f s ℓ 0,1 , s ∈ S(0, 1) are essential bases with respect to the total order (2.4).Since L m,M is defined as cyclic product of L 1,0 (m times) and L 0,1 (M times), we know (see Theorem 1.3) that the Minkowski sum S(m [FFL3] imply that vectors f s ℓ m,M , s ∈ S(m, M ) are linearly independent in L m,M .Thanks to Proposition 2.18 we obtain the first claim of our theorem.To prove the second claim, recall that we've shown in the proof of Proposition 2.18 that relations (2.3) allow to rewrite any monomial in terms of f s ℓ m,M , s ∈ S(m, M ).Hence these are the defining relations.

Geometry
Let us define the main geometric object we are interested in.We consider the Grassmannian Gr(d, n) of d-dimensional subspaces of an n-dimensional vector space.This is a smooth projective algebraic variety of dimension d(n − d).The Grassmannian admits an affine paving by Bruhat cells; in particular, there is natural exponential map ı : A d(n−d) → Gr(d, n) from the affine space to the Grassmannian, whose image is the open cell.We extend the map ı to the birational map P d(n−d) → Gr(d, n) and denote by G(d, n) the closure of its graph (here we consider A d (n−d) as an open cell of P d(n−d) ).More precisely, we have the following definition: Using Plücker embedding, one sees that the graph G(d, n) is naturally embedded into the product of projective spaces P d(n−d) × P(Λ d (L)), where L is the n-dimensional vector space with basis ℓ i , i ∈ [n].Let X I = X i 1 ,...,i k be the homogenenous coordinates on P(Λ d (L)) and Y 0 , Y i,j , 1 ≤ i ≤ d ≤ j < n be the homogenenous coordinates on P d (n−d) .Then the corresponding homogeneous coordinate ring H of G(d, n) ⊂ P d(n−d) ×P(Λ d (L)) is the quotient of the polynomial ring in variables X I , Y 0 , Y i,j by the bi-homogeneous ideal , where m is the total degree with respect to the X-variables and M is the total degree with respect to the Y -variables.One of our goals is to describe the spaces H m,M .
3.1.The group action.Recall the Lie algebra sl n contains a normal abelian subgroup N d and a parabolic subgroup P d .
The following observation is simple, but crucial in our approach.Recall the cyclic vector ℓ m,M ∈ L m,M .In what follows we denote by [ℓ m,M ] ∈ P(L m,M ) the line containing the cyclic vector.We adjust the map ı and the chosen basis ℓ 1 , . . ., ℓ n of L in such a way that X 0 is exactly the image of the map ı : Recall the abelian Lie algebra a d (abelian unipotent radicalin sl n ) and its cyclic module V , which is the direct sum of a one-dimensional space V (0) (spanned by the cyclic vector) and the space V (1) isomorphic to a d as a vector space.In particular, by definition G(d, n) Then the value of Y i,j on exp(a)ℓ 0,1 and the value of X [d]\{i}∪{j+1} are equal to a i,j (note that Y 0 and X Then we claim that Y 0 also vanishes on a point of G(d, n).
Let us show that actually the condition Y 0 = 0 implies vanishing of all the Plücker coordinates X I .In fact, assume Then among Plücker relations one has (see e.g.[Fu]) and hence the following polynomial vanishes on G By induction on the cardinality of B we may assume that the right hand side vanishes.Hence if Y 0 = 0, then all X I = 0, which is not possible.
To conclude the proof we note that the zero set of Y 0 in P(V ) is P(V (1)) and the zero set of X Let us fix the decomposition L = L − ⊕ L + , where Then V (1) and a d are naturally identified with the space Hom(L − , L + ) of linear maps from L − to L + .Hence Lemma 3.9 gives a natural embedding Let pr − be the projection from L to L − along L + .Recall the projection ϕ : G(d, n) → Gr(d, n).For a vector f we denote by [f ] = span{f } the line in the corresponding projective space.
Theorem 3.10.For U ∈ Gr(d, n) the preimage ϕ −1 (U ) is a projective subspace in P(Hom(L − , L + )).A pair ([f ], U ) belongs to ϕ −1 (U ) for a linear map f if and only if the following conditions hold: We first prove this theorem for U being a span of basis vectors ℓ i and then deduce the general case.Let us fix J ∈ [n]  d and let U J = span{ℓ j , j ∈ J}.Proposition 3.11.
Proof.We first show that if f satisfies conditions (3.1), then ([f ], U J ) ∈ ϕ −1 (U J ).Our assumption implies the existence of the decomposition for some coefficients z b,a .Let us take the following family of points p(t) ∈ P(V ) × Gr(d, n) (recall that v ∈ V is the spanning vector of V (0)): Since the preimage of a point is closed, the condition det z b,a = 0 can be omitted.Now let us show that if ( and show that z b,a = 0 unless a ∈ [d] \ J ≤d and b ∈ J >d . U J is of the desired form (with U = U J ).We have the decomposition This proves the desired claim.
In particular, the projection ϕ : To prove the "in particular" part we note that X 0 is the open dense cell and hence ϕ is a bijection on X 0 by definition.
Example 3.14.Let d = 2 with an arbitrary n.Then Gr(2, n) = X 0 ⊔X 1 ⊔X 2 and X 2 ≃ Gr(2, n − 2).The preimage over a point from X 2 is isomorphic to P 3 and all other fibers are just points.Proof.We note that the codimension of X k in the Grassmannian Gr(d, n) is equal to k 2 .In fact, Now it suffices to note that for U ∈ X k the preimage of U is the projective space of dimension k 2 − 1.
Yet another consequence of Theorem 1.3 and Corollary 1.4 (also via [FFL3]) is the following corollary.Proof.Let B be an arbitrary square submatrix of A A ′ and let Q ⊆ P and Q ′ ⊆ P be the rows of A and A ′ , respectively, that appear in B. Let B ′ be the matrix obtained from B by the following row operations: For every β ∈ Q ∩ Q ′ , subtract A ′ β from the row A β .Note that every column of B ′ contains at most two nonzero entries.Moreover, if a column has two nonzero entries, then these are 1 and −1.Observe that this implies that | det B ′ | ∈ {0, 1}.Indeed, if every column of B ′ has exactly two nonzero entries, then the rows of B ′ sum to the zero vector.Othrewise, expanding det B ′ in a column with at most one nonzero entry shows that | det B ′ | is either zero or equals | det B ′′ | for some matrix B ′′ whose columns have the same property as the columns of B ′ .Duality of linear programming and the well-known fact that total unimodularity of the constraint matrix of a linear program guarantees that its optimal value is attained on some integer vector (see [Sch,Chapter 19]) imply that the maximum of (P) equals the minimum of the following integer version of its dual (D):  Suppose that the common value of (P) and (D) is nonzero and let g ∈ Z P ≥0 and h ∈ Z P be two feasible vectors achieving the minimum of (D).Choose an arbitrary δ ∈ P with g δ ≥ 1.It is straightforward to verify that g − e δ and h are a feasible solution to (D) with z replaced by z − e δ .Since clearly β∈P (g − e δ ) β = β∈P g β − 1, the we may use Claim A.2.1 to conclude that M (z − e δ ) ≤ M (z) − 1, as claimed.
d ⋉ P d .We denote the Lie algebra of SL

n
and the (abelian) Lie algebra of N d by a d .Construction 0.2.For each pair m, M ∈ Z >0 there exists an sl (d) n module L m,M , which is cyclic with respect to the action of a d such that G(d, n) embeds into P(L m,M ) as the closure of the N d orbit through the cyclic line.
is isomorphic to a d ⊕ p d as a vector space.The Lie algebra structure on sl(d)n is defined as follows: p d is a subalgebra, a d is an abelian ideal and the adjoint action of p d on a d is induced by the identification of a d with the natural p d module sl n /p d .

n
action by letting p d act on V M (r) as on S r (sl n /p d ).

Definition 2. 8 .
For m, M ≥ 0 we define the a d module L m,M = grL mω d ⊙ V M .Lemma 2.9.L m,M are cyclic a d modules, L m,M = grL ⊙m ω d ⊙ V ⊙M .The a d action extends to the action of sl (d) n .Proof.The first claim is a reformulation of the definition.The second claim holds true since all the factors are sl (d) n modules.In what follows we denote by ℓ m,M ∈ L m,M the cyclic vector; ℓ m,M = ℓ mω d ⊗ v M .Remark 2.10.We show in the next section that the spaces L m,M are closely related to the closure in P d(n−d) × Gr(d, n) of the graph G(d, n) of the birational exponential map P d(n−d) → Gr(d, n).
15.Our goal is to show that relations (2.3) are defining relations of L m,M as sl (d) n module.However, they are not defining if L m,M is considered only as an a d module.In other words, the defining relations of the a d module L m,M are obtained from (2.3) applying the universal enveloping algebra U(p d ).

n
= p d ⊕ a d and its representations L m,M .Let P d ⊂ SL n be the standard parabolic subgroup with the Lie algebra p d and let N d ⊂ SL n be the unipotent subgroup with the Lie algebra a d .Recall that a d is endowed with the structure of p d module and hence of the P d module.Hence one obtains the group the open dense cell A d(n−d) ⊂ P d(n−d) and by definition [(ℓ 1 + aℓ 1 ) ∧ • • • ∧ (ℓ d + aℓ d )] = ı(a) (the left hand side is the standard parametrization of the open cell in the Grassmannian).Hence G 1,1 ≃ G(d, n).Corollary 3.4.The abelian unipotent group N d ≃ G d(n−d) a acts on G(d, n) with an open dense orbit.The N d action on G(d, n) extends to the action of the group SL (d) n .3.2.The fibers.By definition we have a projection map ϕ : G(d, n) → Gr(d, n).Our goal is to describe the fibers of the map ϕ.To this end, we consider the following stratification of the Grassmannians.Recall that we have fixed n-dimensional space L, the points of Gr(d, n) are d dimensional subspaces of L. Definition 3.5.Let X k ⊂ Gr(d, n) be the set of subspaces U ⊂ L such that dim (U ∩ span{ℓ d+1 , . . ., ℓ n }) = k.By definition, X k is nonempty if and only if k ≤ min(d, n − d).Remark 3.6.Each X k is a union of several Schubert cells in the Grassmannian.More precisely, the Schubert cells C I ⊂ Gr(d, n) are labeled by subsets I ⊂ [n] such that |I| = d.Then X k is the union of the cells C I such that |I ∩ {d + 1, . . ., n}| = k.In particular, X 0 is the open cell.

3. 3 .
Toric degenerations and coordinate rings.In this subsection we collect the geometric consequences from Theorem 1.3 and Corollary 1.4.Corollary 3.18.The homogeneous components H m,M of the homogeneous coordinate ring H of G(d, n) are isomorphic to L * m,M as sl (d) n modules.