SYMPLECTIC PBW DEGENERATE FLAG VARIETIES; PBW TABLEAUX AND DEFINING EQUATIONS

We define a set of PBW-semistandard tableaux that is in a weight-preserving bijection with the set of monomials corresponding to integral points in the Feigin–Fourier–Littelmann–Vinberg polytope for highest weight modules of the symplectic Lie algebra. We then show that these tableaux parametrize bases of the multihomogeneous coordinate rings of the complete symplectic original and PBW degenerate flag varieties. From this construction, we provide explicit degenerate relations that generate the defining ideal of the PBW degenerate variety with respect to the Plücker embedding. These relations consist of type Α degenerate Plücker relations and a set of degenerate linear relations that we obtain from De Concini’s linear relations.


Introduction
Let G be a simple, simply connected algebraic group over the field C and g the corresponding Lie algebra.Let g = n + ⊕ h ⊕ n − be a Cartan decomposition and b = n + ⊕ h the Borel subalgebra.For a dominant, integral weight λ, let V λ be the corresponding simple g-module, and ν λ ∈ V λ a highest weight vector.For λ regular, the complete flag variety F λ is defined to be the closure of the G-orbit through a highest weight line: F λ = G[ν λ ] ֒→ P(V λ ).Another realisation of this variety is through the quotient G/B, where B is a Borel subgroup.
On the other hand, one has V λ = U (n − )ν λ , where U (n − ) is the universal enveloping algebra of n − .There exists a degree filtration U (n − ) s = span{x 1 • • • x l : x i ∈ n − , l ≤ s} on U (n − ).This filtration in turn induces the filtration F s = U (n − ) s ν λ on V λ , called the PBW filtration.The associated graded space is F 0 ⊕ s≥1 F s /F s−1 , which will be denoted by V a λ (see [FFL1] and [FFL2]).This graded space has a structure of g a -module where g a is a Lie algebra which is a semi-direct sum of b and an abelian ideal (n − ) a .Let G a be a Lie group corresponding to g a .Let ν a λ be the image of ν λ in V a λ .The PBW degenerate flag variety is defined to be F a λ := G a [ν a λ ] ֒→ P(V a λ ) ( [FEI]).
Feigin in [FEI], studied the variety F a λ in type A when G = SL n (C) and g = sl n (C).The Plücker embedding of the original variety F λ and the PBW degenerate variety F a λ into the product of projective spaces was considered, i.e., In order to show that the variety F a λ is a flat degeneration of the original variety F λ , he defined the PBW-semistandard tableaux which label bases of the multi-homogeneous coordinate rings of both varieties.Let us review what these tableaux are.For a type A n−1 dominant, integral weight λ, written as a partition λ = (λ 1 ≥ λ 2 ≥ • • • ≥ λ n−1 ≥ 0), consider the corresponding Young diagram Y λ (English convention).
A type A PBW-semistandard tableau of shape λ is the filling of Y λ with entries from {1, . . ., n} such that the following three conditions are satisfied.First of all, in each column, each entry less than the length of that column is at row position equal to that entry (or in short, at its position).Secondly, every entry that is not at its position should be greater than all entries below it in any given column.And finally, for every entry in each column apart from the first column, there should be a greater or equal entry in the column to the left and in the same row or in a row below.We refer to the last condition as PBW-semistandardness. Now consider type C, with G = Sp 2n (C) and g = sp 2n (C).We consider the complete symplectic flag variety, which will be denoted by SpF 2n and its PBW degeneration, which will be denoted by SpF a 2n .We again consider the Plücker embedding of these varieties into the product of projective spaces: and SpF a 2n ֒− → Let C[SpF 2n ] and C[SpF a 2n ] denote the multi-homogeneous coordinate rings of SpF 2n and SpF a 2n with respect to the above embeddings.The first goal of this paper is to define a set of PBW-semistandard tableaux for type C n , and to show that it labels weighted bases of both C[SpF 2n ] and C[SpF a 2n ].Let λ be a type C n dominant, integral weight, written again as a partition λ = (λ 1 ≥ λ 2 ≥ • • • ≥ λ n ≥ 0).For i ∈ {1, . . ., n}, let i := 2n + 1 − i.
A symplectic (or type C) PBW-semistandard tableau of shape λ, is a filling of the corresponding Young diagram Y λ with entries in the set {1 < • • • < n < n < • • • < 1} such that not only the conditions for the type A PBW-semistandard tableaux are satisfied, but also the following extra condition.For every element i ∈ {1, . . ., n} in any column, if the element i exists in the same column, then the position of i should be above that of i, whenever i is less than the length of the column.We would like to note that several nice symplectic tableaux already exist, for example those of De Concini [DEC], Hamel and King [HK], Kashiwara and Nakashima [KN], King [KIN], and Proctor [PRO].The main difference between these tableaux and those defined here is the PBW-semistandardness condition (see Subsection 3.4 for a brief comparison).We prove Theorem 1.1 (Theorem 5.9).The symplectic PBW-semistandard tableaux index a basis of C[SpF a 2n ].Feigin, Finkelberg and Littelmann showed in [FFIL] that SpF a 2n is a flat degeneration of SpF 2n .It therefore follows naturally that the symplectic PBW-semistandard tableaux also label a basis for C[SpF 2n ] (see Proposition 4.16).In the light of their combinatorics, we would like to discuss a correspondence between these tableaux and certain bases of the modules V λ and V a λ .In [FFL1] and [FFL2], Feigin, Fourier and Littelmann defined the Feigin-Fourier-Littelmann-Vinberg polytopes that parametrize monomial bases for highest weight original and PBW degenerate simple modules for a Lie algebra g in types A and C respectively.Bases arising this way are called FFLV bases.Their existence was first conjectured by Vinberg (see [VIN]), who also proved the conjecture for sl 4 , sp 4 and G 2 .We prove that one has a weight preserving bijection between the FFLV basis for the symplectic modules V λ and V a λ and the symplectic PBW-semistandard tableaux (see Theorem 3.16).It is worth noting that Young [YNG] was the first to introduce (semi-)standard Young tableaux to provide a basis for the irreducible polynomial representations of the general linear group and for the irreducible representations of symmetric groups.On the other hand, standard monomial theory was begun by Hodge [HOD], who used Young theory to give a basis for homogeneous coordinate rings of Grassmann and flag varieties.The same theory has been tremendously developed through the work of different authors ( [DEC], [LMS], [LS], [LIT], . ..).
At this point, we would like to step back and discuss briefly one of the very important tools in the proof of Theorem 5.9; namely, the degenerate relations.Feigin in [FEI] defined the degenerate Plücker relations and proved that they generate the defining ideal of the PBW degenerate flag variety in type A. Since SpF a 2n is point-wise contained in the type A 2n−1 complete PBW degenerate flag variety ( [FFIL]), it follows that Feigin's degenerate relations are also satisfied on SpF a 2n .We denote these relations by R t;a L,J .
On the other hand, De Concini [DEC] defined certain linear relations while showing that his symplectic standard tableaux index a basis for C[SpF 2n ].We call these symplectic relations, which will be denoted by S (I 2 ,I 1 ) .In his proof, he also used Plücker relations, which implies that these and the symplectic relations generate the defining ideal of SpF 2n , since they provide a straightening law for C[SpF 2n ].Note that Chirivì and Maffei in [CM] and in [CML] with Littelmann, gave a general framework for these defining equations for flag varieties and other spherical varieties.We now obtain degenerate relations from the symplectic relations, which we call symplectic degenerate relations and denote them by S a (I 2 ,I 1 ) (see Definition 5.3 for a full description).We obtain a fundamental result about the defining ideal of SpF a 2n , which is the second and final goal of this paper.Let I a be the ideal generated by the relations S a (I 2 ,I 1 ) and R t;a L,J .For example, for n = 2, the ideal I a is generated by the relations: We prove: Theorem 1.2 (Theorem 5.10).The ideal I a is the prime defining ideal of the variety SpF a 2n under the Plücker embedding, In the framework of PBW degenerations, the root vectors of the Lie algebra n − are each assigned degree one.It is interesting to note that one can systematically assign different weights to these root vectors such that the associated graded space of n − still naturally admits a non-trivial graded Lie algebra structure.To such a structure, one then associates an appropriate Lie group structure.Weighted analogues of the usual PBW degenerate flag varieties arise this way.
This construction was carried out by Fang, Feigin, Fourier and Makhlin in [FFFM] for type A. They used the combinatorics of PBW-semistandard tableaux to show that these varieties are well behaved, in the sense that they are irreducible.In the sequel, they constructed an explicit maximal prime cone of the tropical flag variety coming from the system of weights assigned to the root vectors of n − , hence obtaining yet another phenomenal connection of Lie theory to tropical geometry.Moreover, they showed that every point in the relative interior of this cone corresponds to the FFLV toric degeneration.Furthermore, they identified several facets corresponding to linear degenerations ( [CFFFR]).
In a forthcoming work, we will carry out similar constructions for sp 2n and hence establish connections to the tropical symplectic flag variety ( [BO]).In the same spirit, we are also extending the work of Bossinger, Lambogila, Mincheva and Mohammadi [BLMM] on computing toric degenerations arising from tropicalization of flag varieties to the symplectic set-up.This paper is organised as follows: In Section 2, we recall results on the FFLV basis for the symplectic Lie algebra.In Section 3, we define the symplectic PBW-semistandard tableaux and establish the bijection between them and the symplectic FFLV basis.We then show that these tableaux label a basis for the multi-homogeneous coordinate ring of SpF 2n in Section 4. In Section 5, we give the definition of the symplectic degenerate relations and use them together with the degenerate Plücker relations to show that the symplectic PBW-semistandard tableaux label a basis for the coordinate ring of SpF a 2n .We also prove here that the ideal generated by these relations is the defining ideal of SpF a 2n under the Plücker embedding.
Acknowledgements.The author would like to extend his gratitude to his doctoral advisor, Ghislain Fourier, for many useful and insightful discussions on this work and its extensions.Many thanks to anonymous referees for key comments and suggestions that improved on the presentation of this paper.Similarly, great thanks to Xin Fang, Evgeny Feigin, Johannes Flake, Peter Littelmann, and Jorge Alberto Olarte, for several important expositions on these subjects.The author also extends his gratitude to Johannes Flake for technical support with the computer codes that verified our results and to Xin Fang for reading the first version of this paper.The author is funded by the Deutscher Akademischer Austauschdienst (DAAD, German Academic Exchange Service) scholarship: Research Grants -Doctoral Programmes in Germany (Programme-ID 57440921).This work is a contribution to Project-ID 286237555 -TRR 195 --by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

Preliminaries; Representation Theory
In this section, we recall the description of the corresponding simple original and PBW degenerate modules for the symplectic Lie algebra and the FFLV basis as studied in [FFL2].
2.1.The Symplectic Lie Algebra; a Brief Description.All information in this subsection can be found in [FH].Let g = sp 2n (C) = n + ⊕ h ⊕ n − be a Cartan decomposition and b = n + ⊕ h the Borel subalgebra.Let Λ + denote the set of dominant integral weights of sp 2n , and let {ω 1 , . . ., ω n } be the set of fundamental weights.Let {ε 1 , . . ., ε n } be the standard basis for h * with respect to the killing form.Let α 1 , . . ., α n be the simple roots of sp 2n and let Φ + denote the set of positive roots.For 1 ≤ j ≤ n, let j := 2n + 1 − j.The set Φ + is the union of the following two sets of roots: where α i,n = α i,n .For each α ∈ Φ + , fix a non zero element f α ∈ n − −α , where n − −α is the set of root vectors in n − of weight −α.Henceforth, we will sometimes use the short forms: Let W be a 2n-dimensional vector space over C and let {w 1 , . . ., w 2n } be its fixed basis.We fix a non-degenerate symplectic form , , defined by: w i , w i = 1 for 1 ≤ i ≤ n and w i , w j = 0 for all 1 ≤ i, j ≤ n, j = i.The symplectic group Sp 2n (C) can be realized as the group of automorphisms of W leaving the form , invariant.Consider a maximal torus T ⊂ Sp 2n consisting of diagonal matrices t given as follows: , and a Borel subgroup B ⊂ Sp 2n of upper triangular matrices.With respect to this realization, explicit formulas for root vectors of the symplectic Lie algebra sp 2n are given below: where E p,q is the matrix with zeros everywhere except for the entry 1 in the p-th row and q-th column.
2.2.The Poincaré-Birkhoff-Witt (PBW) Degeneration.Consider the increasing degree filtration on the universal enveloping algebra, U (n − ): For a dominant integral weight λ = m 1 ω 1 + . . .+ m n ω n ∈ Λ + , let as usual, V λ be the corresponding simple highest weight sp 2n -module with a highest weight vector ν λ .It is known that V λ = U (n − )ν λ , therefore, the filtration (2.2) induces an increasing degree filtration F s on V λ : This filtration is called the PBW filtration.Let us denote the associated graded space by V a λ .One has: Elements of V a λ (s) are said to be homogeneous of PBW-degree s.The graded space V a λ has a structure of g a -module, where g a is a semi-direct sum of the Borel subalgebra b and an abelian ideal (n − ) a , which is isomorphic to n − as a vector space.The Lie algebra g a is said to be the PBW degeneration of g (see [FEI]).For the highest weight vector ν λ in V λ , we denote by ν a λ its image in V a λ .2.3.The Symplectic FFLV Basis.Here we recall results due to Feigin,Fourier and Littelmann in [FFL2].Our results on the symplectic PBW-semistandard tableaux strongly rely on these results.We first recall the notion of the symplectic Dyck path.Let J denote the set {1, . . ., n, n − 1, . . ., 1} with the usual order: of positive roots satisfying the conditions: (i) the first root d(0) = α i for some 1 ≤ i ≤ n, i.e., it is simple.
(ii) the last root is either simple or the highest root of a symplectic subalgebra, i.e., d(k) = α j or d(k) = α j for some 1 ≤ j < n. (iii) the elements in between satisfy the recursion rule: If d(s) = α p,q with p, q ∈ J, then the next element in the sequence is of the form either d(s + 1) = α p,q+1 or d(s + 1) = α p+1,q ; where x + 1 denotes the smallest element in J which is bigger than x.
Example 2.2.For sp 6 ; the roots can be arranged in form of a triangle as shown below.The Dyck paths are the ones starting at a simple root and ending at one of the edges following the directions indicated by the arrows.
Definition 2.3.Denote by D the set of all Dyck paths.For a dominant, integral weight λ = n i=1 m i ω i ∈ Λ + , the symplectic Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope is the polytope of the points p = (p α ) α>0 ∈ R n 2 ≥0 such that for all d ∈ D, the following inequalities are satisfied: In what follows, we will sometimes also use the shorthand notation p i,j = p α i,j and p i,j = p α i,j .
Let S(λ) be the set of integral points in P(λ).For a multi-exponent p = (p α ) α>0 , p α ∈ Z ≥0 , let f p be the monomial element: where S(n − ) denotes the symmetric algebra of n − .
Theorem 2.5 ( [FFL2]).The elements {f p ν a λ , p ∈ S(λ)} form a basis of V a λ and {f p ν λ , p ∈ S(λ)} form a basis of V λ (after fixing a total order for the root vectors in each monomial f p ).
In what follows, we will refer to the basis {f p ν λ , p ∈ S(λ)} as the symplectic FFLV basis.For any two dominant integral weights λ and µ, one has a unique injective homomorphism of modules, We end this section by stating an analogous result in the PBW degenerate case: FFL2]).For any two dominant, integral weights λ and µ, there exists an injective homomorphism of modules:

The [Symplectic FFLV Basis] -[PBW Tableaux] Correspondence
In this section, we define a set of PBW-semistandard tableaux that is in a one-to-one correspondence with the symplectic FFLV basis.We explicitly construct the corresponding maps, first for fundamental weights and then we later generalise to any dominant, integral weight.These tableaux take entries in the set N := {1, . . ., n, n, . . ., 1}, with the usual order: 1 Before we proceed, we first recall a few preliminary notions on Young diagrams and tableaux: 3.1.Young Diagrams and Tableaux.Given a partition λ = (λ 1 ≥ λ 2 ≥ . . .≥ λ n ≥ 0), the corresponding Young diagram, which we denote by Y λ , is a finite collection of boxes arranged in left-justified rows.The rows and columns in Y λ are numbered from top to bottom and left to right respectively.Therefore, to every box of Y λ we assign a pair (i, j) for 1 ≤ i ≤ n and 1 ≤ j ≤ λ 1 .For example, Y (6,4,4,2) is the following diagram: A Young tableau T λ is a filling of Y λ with numbers T i,j ∈ N , where T i,j denotes the number put in the box labelled by the pair (i, j).We call the partition λ the shape of the tableau T λ .A tableau T λ is called a semistandard Young tableau if the entries T i,j are such that they are strictly increasing down the columns and weakly increasing across the rows from left to right.The tableaux we define below are analogues of the semistandard Young tableaux.
Finally, to a dominant, integral weight λ = n k=1 m k ω k , we assign a partition (m which we label by the same symbol λ.Moreover, from a given Young diagram Y λ , we can recover the weight λ by associating a fundamental weight ω k to each column of length k, and then summing up.For example, the weight corresponding to the Young diagram Y (6,4,4,2) shown above is λ = 2ω 1 + 2ω 3 + 2ω 4 .
3.2.The Case of Fundamental Weights.In this subsection, we set λ to be a fundamental weight, i.e., λ = ω k for 1 ≤ k ≤ n.To such a weight, we associate as described above, a partition λ = (1, . . ., 1) k−times . (3.1) The Young diagram of such a partition is just a single column of length k.We have the following definition: Definition 3.1.For a partition λ given in (3.1) above, the symplectic PBW tableau T λ is the filling of the corresponding Young diagram Y λ with numbers T i ∈ N such that: (in this case we say that the entry T i is at its position), (ii) if i 1 < i 2 and T i 1 = i 1 , then T i 1 > T i 2 , and (iii) if there exist i, i ′ with T i = i and T i ′ = i, then i ′ < i, whenever i < k.
Let SyP λ be the set of all elements f p • ν λ with the product f p given as in Equation (2.4).Recall that SyP λ is the symplectic FFLV basis for the sp 2n -module V λ .Also, let SyT λ denote the set of all symplectic PBW tableaux of the shape λ that is given in (3.1).We want to establish a weight preserving bijection between SyP λ for λ = ω k and SyT λ .To do this, we first describe in the following definition, how to assign a symplectic PBW tableau to each element of SyP λ .Definition 3.3.Let λ = ω k be a fundamental weight for 1 ≤ k ≤ n.Let t λ denote the highest weight single column tableau, i.e., the tableau with u appearing in box u for all 1 ≤ u ≤ k.To an element f p • ν λ , we assign an element f p • t λ by applying each operator f u,v appearing in f p to entry u of t λ according to the following rule: where n + 1 = n.
In the following example we illustrate the above rule: Example 3.4.Consider λ = ω 3 and n = 3.We will describe how the fifth tableau in the list of tableaux in Example 3.2 is obtained by the above rule.For this, consider the element f 2,2 f 3,3 • ν ω 3 in the symplectic FFLV basis for the sp 6 -module V ω 3 .One has: Recall the basis {ε 1 , . . ., ε n } for h * .Let wt(z) denote the weight of an object z (for example, tableau, highest weight vector, etc).In the following definition, we assign weights to the different objects we are dealing with.Definition 3.5.Let λ = ω k be a fundamental weight.Let T λ be a symplectic PBW tableau of shape λ, N + := {i ∈ {1, . . ., n} : i ∈ T λ } and N − := {j ∈ {1, . . ., n} : j ∈ T λ }.The weight of T λ is given by: The weights of the root vectors f i,j and f i,j are their usual weights in the Lie algebra n − , i.e., Remark 3.6.From the above definition, it follows that for the product f p = α>0 f p α α , the weight is: We also have wt(f p • t λ ) = wt(f p ) + wt(t λ ) and wt(t λ ) = wt(ν λ ).Notice that the weight of the element f p • ν λ as given here is its actual weight in the module V λ .
We prove the following result: Proposition 3.8.For λ = ω k a fundamental weight, the set SyP λ is in a weight preserving one-to-one correspondence with the set SyT λ .
Proof.Define the assignment: where f p • t λ is given according to Definition 3.3.
We will show that θ 1 is a well defined map.Since the elements of the set SyP λ form a basis for the sp 2n -module V λ for λ = ω k according to Theorem 2.5, it suffices to show that for any . ., α is,js be the roots corresponding to the root vectors appearing in f p • ν λ .Now since λ = ω k is a fundamental weight, then according to Definition 2.3, all inequalities are of the following form: . . .+ p i l ,j l + . . .≤ 1, for 1 ≤ l ≤ s.This implies that the roots α i 1 ,j 1 , . . ., α is,js are not pairwise on a common Dyck path.This in turn means that i 1 = . . .= i s , since at least two roots would lie on the same Dyck path otherwise.Now pairs of roots that don't lie on the same Dyck path are of the form: (i) α p,q and α p+1,q−1 which implies p < p + 1 and q > q − 1, or (ii) α p,q and α p−1,q+1 which implies p − 1 < p and q + 1 > q, where x + 1 is an element in J = {1, . . ., n, n − 1, . . ., 1} that is bigger than x while x − 1 is an element in J that is smaller than x.Since this is true for all pairs of roots and we have 1 . ., f is,js each act on a different entry of the single column highest weight tableau t λ once.Let We note that the entries of t λ which are not acted upon remain at their positions, so condition (i) of Definition 3.1 is satisfied.The elements i ′ 1 , . . ., i ′ s are the ones that are not at their positions.Condition (ii) is therefore satisfied since we have i We are left with showing that condition (iii) holds true.For an entry m in the tableau f p • t λ with m < k, we need to check that if m exists in the tableau, then its position is above that of m.Consider ), so m is above m, and we are done.
We also define the assignment: by associating a root vector f i l ,j l to an element of T λ that is not at its position for all l with 1 ≤ l ≤ s.Let x 1 , . . ., x s denote the entries of T λ that are not at their positions with . ., i s denote the box numbers of the entries x 1 , . . ., x s respectively.Then the operator f i l ,j l for 1 ≤ l ≤ s is obtained by the following rule: We claim that θ 2 is a well defined map.For this, we will show that θ We have 1 ≥ j 1 > • • • > j s > k, since the elements x 1 , . . ., x s are not at their positions in T λ .We also have 1 Therefore, for 1 ≤ l ≤ s, each root α i l ,j l corresponding to the root vector f i l ,j l lies in some Dyck path with no two distinct roots lying in a common Dyck path.So, the point (. . ., p i l ,j l , . ..) with p i l ,j l = 1 satisfies an inequality of the form: , where id denotes the identity map.Consider 2), we have: ) with the entries x 1 , . . ., x s of T λ obtained from f i l ,j l for 1 ≤ l ≤ s according to (3.2).Now applying θ 2 to T λ , we get: which means that the maps θ 1 and θ 2 are inverse to each other.So, we have constructed the required bijection.
We are now left with proving that the defined maps are weight preserving.For this, it suffices to show that the map θ 1 is weight preserving, i.e., that wt(θ 1 (f p • ν λ )) = wt(f p • ν λ ).Indeed we have: 3.3.The Case of Dominant Integral Weights.In this subsection, we extend results from the previous subsection on the case of fundamental weights to the case of any dominant integral weight λ = n k=1 m k ω k of sp 2n .As before, we will denote by λ the corresponding partition, i.e., λ Definition 3.9.For a dominant integral weight λ, a symplectic PBW tableau T λ whose shape is the corresponding partition λ, is a filling of the corresponding Young diagram Y λ with numbers T i,j ∈ N such that for µ j , the length of the j-th column, we have: We say that a symplectic PBW tableau T λ is PBW-semistandard if in addition, the following condition is satisfied: (iv) for every j > 1 and every i, Example 3.10.For n = 2, and λ = (2, 1) (i.e., λ = ω 1 + ω 2 ), the set of all the 16 symplectic PBW-semistandard tableaux is the one given below: Denote by SyST λ the set of all symplectic PBW-semistandard tableaux of shape λ on the set N as above.
We introduce a total order on the operators f i,j , f i,j as follows: We say f i 1 ,j 1 > f i 2 ,j 2 if either i 1 < i 2 or i 1 = i 2 and j 1 < j 2 .For example, we have f 11 > f 22 and f 12 > f 11 .We now order our operators in the product f p = α>0 f p α α according to this total order.
In the following definition, we extend the assignment described in Definition 3.3 to the case of dominant integral weights.Definition 3.11.Let λ be a dominant integral weight and t λ be a highest weight tableau, i.e., a tableau with one's in the first row, two's in the second row, and so on.We define the assignment f p • t λ as follows.Apply the operators in the ordered product f p starting with the smallest one.An operator f i,j acts on entry i in column c whenever j ≥ µ c , where c is the first column from the left where this is true.The assignment f p • t λ then narrows down to the assignment f i,j • i of each operator f i,j in the product f p only once on the entry i in the best choice column c of t λ according to rule (3.2) in Definition 3.3.
Example 3.12.For sp 4 and λ = ω 1 + ω 2 , one has 16 integral points of the polytope P(λ).This leads to the following set of monomials: 11 f 22 }, each of them corresponding to the symplectic PBW-semistandard tableau appearing in the same position in the list of tableaux given in Example 3.10.For an illustration of how the assignment described in Definition 3.11 works, consider the second last monomial in the list above.Then one has: The resulting tableau is the second last one in the list of tableaux in Example 3.10.
Proposition 3.13.The assignment where f p • t λ is given according to Definition 3.11 is a well defined map.
Proof.Since the elements of the set SyP λ are a basis for the sp 2n -module V λ according to Theorem 2.5, it suffices to show that for an arbitrary element We begin 'acting' with the smallest operator f is,js in the first column c 1 from the left for which j s ≥ µ c 1 .We then proceed to the next smallest one js be the product of the operators which act in the same column.The result of this product satisfies all conditions of symplectic PBW tableaux defined on columns according to Proposition 3.8.Now let f i s−k−1 ,j s−k−1 be the next smallest operator for which i s−k−1 ≤ i s−k and j s−k−1 ≤ j s−k .This operator acts in the next column c 2 to the right of the column c 1 .Because of the above argument, it suffices to check that the two columns c 1 and c 2 satisfy condition (iv) of Definition 3.9. If So under the assignment φ, we have: We thus have the entry j s−k + 1 in row i s−k and column c 1 and the entry j s−k−1 in row i s−k−1 and column c 2 , such that j s−k−1 + 1 ≤ j s−k + 1, which implies that condition (iv) of Definition 3.9 is satisfied in this case.In like manner, we check the other cases as follows: Here we have two cases: Hence condition (iv) of Definition 3.9 is again satisfied following the above argument.(ii) if n − 1 ≤ j s−k−1 ≤ 1, then we have: Hence we again have i s−k−1 ≤ i s−k and j s−k−1 ≤ j s−k .Therefore condition (iv) of Definition 3.9 still holds true.We continue in the same way until all the operators are applied.
We now construct the inverse map to the map φ described in Proposition 3.13.To do this, we describe in the following definition how to recover an element f p • ν λ from a tableaux T λ ∈ SyST λ .Definition 3.14.Given a tableaux T λ ∈ SyST λ , let h denote the entry in the box labelled by the pair (r, c), where r is the row number and c is the column number.Let µ c denote the length of the column c.An element f p • ν λ is obtained from T λ in the following way.
(a) To each entry h in T λ that is greater than µ c , apply the following assignment: where n − 1 = n.(b) Let h ′ denote h or h − 1 as the case may be after applying the above assignment.We put together the operators f r,h−1 and f r,h to obtain the product The weight λ is obtained from the shape of the tableau T λ as described in Subsection 3.1.
This and step (b) above yield the element f p • ν λ .
We prove: Proposition 3.15.The assignment where f p • ν λ is obtained according to Definition 3.14 is a well defined map.
Proof.For any To prove this, we proceed as follows.Consider any two arbitrary neighbouring columns j 1 and j 2 in T λ .Let µ j 1 = l and µ j 2 = s such that 1 ≤ s ≤ l ≤ n.Let {x 1 , . . ., x l } be elements from j 1 and {y 1 , . . ., y s } be elements from j 2 .It suffices to consider only those elements that are not at their positions.Let {x t 1 , . . ., x t k } be elements from j 1 that are not at their positions and likewise {y r 1 , . . ., y r k } be elements from j 2 that are not at their positions with 1 According to the definition of a symplectic PBW-semistandard tableau, we have that {x We put the elements in j 1 and j 2 together and arrange them in descending order.Let {x t 1 , . . ., x t z−1 } be the first z−1 elements that lie in column j 1 .Let be the corresponding monomial got by applying the map θ 2 from Proposition 3.8.Now assume the next biggest element y rz lies in j 2 .Then there must exist x t z+1 with t z+1 ≥ r z such that x t z+1 ≥ y rz .
We continue in the same way until all the elements in the columns j 1 and j 2 are done.We now put together all products of the form and all products of the form f i 1 ,j 1 f i 2 ,j 2 to obtain the monomial f p according to Definition 3.14.From the shape λ of the tableau T λ , we recover the highest weight vector ν λ as described in Subsection 3.1.Now from the above argument and from Proposition 3.8, it follows that the element f p • ν λ lies in SyP λ .
We extend the definition of weights from Definition 3.5 to the case of dominant integral weights by considering all columns in the tableaux T λ and t λ , and all operators appearing in the corresponding elements f p • ν λ .We prove: Theorem 3.16.For λ = n k=1 m k ω k a dominant integral weight, the symplectic FFLV basis is in a weight preserving one-to-one correspondence with the set SyST λ of symplectic PBW-semistandard tableaux of shape λ with entries in N .
Proof.For the one-to-one correspondence, it suffices to prove that for the maps φ and π constructed in Propositions 3.13 and 3.15 respectively, we have φ • π = π • φ = id, where id is the identity map.
Let us begin with proving that φ • π = id.We again consider two neighbouring columns j 1 and j 2 with µ j 1 ≥ µ j 2 .It suffices to consider elements that are not at their positions as before.As before, let {x t 1 , . . ., x t z−1 } be elements in the column j 1 , and y rz the next element which is in the column to the right, j 2 , such that ∃ x t z+1 with x t z+1 ≥ y rz and t z+1 ≥ r z .If µ j 1 < x t z+1 ≤ n, then we have φ • π(T λ ) = φ(f rz,yr z −1 f t z+1 ,xt z+1 −1 • ν λ ), so we have: Moreover, we have that f rz,yr z −1 ≥ f t z+1 ,xt z+1 −1 under our total order with equality if and only if r z = t z+1 and y rz − 1 = x t z+1 − 1. Therefore the operator f t z+1 ,xt z+1 −1 acts only in the left-hand column j 1 , since x t z+1 − 1 ≥ µ j 1 and the operator f rz,yr z −1 acts in j 2 since y rz − 1 ≥ µ j 2 .So, we have ), so we have: Again we have that f rz,yr z −1 > f t z+1 ,xt z+1 −1 under our total order.Therefore the operator f t z+1 ,xt z+1 acts only in the left-hand column j 1 , since x t z+1 ≥ µ j 1 and the operator f rz,yr z −1 acts in j 2 since ), so we have: Again we have that f rz,yr z > f t z+1 ,xt z+1 under our total order.Therefore the operator f t z+1 ,xt z+1 acts only in the left-hand column j 1 , since x t z+1 ≥ µ j 1 and the operator f rz,yr z acts in j 2 since y rz ≥ µ j 2 .

So again
js is the product of the operators which act in the same column j 1 .Let f i s−k−1 ,j s−k−1 be the smallest operator for which i s−k−1 ≤ i s−k and j s−k−1 ≤ j s−k .This operator acts in the right-hand column j 2 .If µ j 1 ≤ j s−k ≤ n, then also µ j 2 < j s−k−1 ≤ j s−k ≤ n.So, we have: where here the pair (i, j) means that at position i of a respective column, we have entry j.
Now we are left with showing that this one-to-one correspondence is weight preserving.For this we need to only show that the map: is weight preserving, i.e., that wt(φ(f p • ν λ )) = wt(f p • ν λ ).For this, we have:

3.4.
A Comparison with Other Existing Tableaux.The PBW-semistandard tableaux of type A are defined as follows: ) for any j > 1 and any i there exists i ′ ≥ i such that T i ′ ,j−1 ≥ T i,j .
It follows that a symplectic PBW-semistandard tableau is a PBW-semistandard tableau of type A which satisfies one extra condition on the columns (condition (iii) of Definition 3.9).
Example 3.18.For g of type A 3 , the full set of PBW-semistandard tableaux restricted to λ = ω 1 + ω 2 (λ = (2, 1)) on the set N = {1, 2, 2, 1} is the one given below: When we consider condition (iii) of Definition 3.9, then we have to drop the last four tableaux from the above list.This way, we are able to recover all the 16 PBW-semistandard tableaux corresponding to λ = ω 1 + ω 2 for g of type C 2 as seen in Example 3.10.
As will be seen in the following section, the symplectic standard tableaux of De Concini in [DEC] are different from the symplectic PBW-semistandard tableaux because a different condition is imposed on the columns.Furthermore, the symplectic semistandard tableaux of Hamel and King [HK], King [KIN], Kashiwara and Nakashima [KN] and Proctor [PRO] yield semistandard Young tableaux when restricted to type A n−1 , i.e., if entries are taken from the set {1, . . ., n}.Hence they are different from the symplectic PBW-semistandard tableaux since the restriction of these in the same way does not yield semistandard Young tableaux.Notice however that there exist weight preserving bijections between all these symplectic tableaux.

The Complete Symplectic Flag Variety; Symplectic Relations and a Basis for the Coordinate Ring
In this section, we describe the complete symplectic flag variety together with its defining ideal, and we show that the symplectic PBW-semistandard tableaux label a basis for the multi-homogeneous coordinate ring.4.1.Flag Varieties; a Brief Description.Let G be a simple, simply connected algebraic group over the field C with the corresponding Lie algebra g.As before, we have a Cartan decomposition g = n + ⊕ h ⊕ n − .We know that V λ has a structure of a G-module with highest weight vector ν λ .Hence we have an action of G on the projectivization P(V λ ).The flag variety F λ is the closure of the G-orbit through the highest weight line: Let λ be any dominant integral weight of g.Assume (λ, α ∨ i ) = 0 if and only if f α i belongs to p, the Lie algebra corresponding to P, a parabolic subgroup of G. Then each projective variety F λ is as well isomorphic to the quotient G/P of G by the parabolic subgroup P leaving Cν λ invariant.This is the generalized/partial flag variety.In particular, when λ is regular, the flag variety F λ is isomorphic to G/B, as projective varieties, where B is a Borel subgroup, and this is then called the complete/full flag variety.

4.2.
The Complete Symplectic Flag Variety; General Description.Now we consider G = Sp 2n (C).Recall the 2n-dimensional vector space W over C and the fixed basis {w 1 , . . ., w 2n }.Recall also the non-degenerate symplectic form , , defined by: w i , w i = 1 for 1 ≤ i ≤ n and w i , w j = 0 for all 1 ≤ i, j ≤ n, j = i, where as before, i = 2n + 1 − i.The matrix of this symplectic form is given by where I n is the n × n matrix with 1's along the anti-diagonal and zeros elsewhere.Recall that an isotropic subspace of a symplectic vector space is a subspace on which the symplectic form identically vanishes.For W as above, all the isotropic subspaces have dimension at most n.For 1 ≤ k ≤ n, the symplectic Grassmannian SpGr(k, 2n) is the quotient of Sp 2n by a maximal parabolic subgroup and it is known to coincide with the variety of isotropic k-dimensional subspaces of W. Notice that when we drop the isotropic condition, we recover the usual Grassmannian which will be denoted by Gr(k, 2n).
We consider the flag variety Sp 2n /B, where B is a Borel subgroup.This is the complete symplectic flag variety which we denote by SpF 2n and it coincides with the variety whose points are the full flags This variety is also referred to as the isotropic flag variety as in [DEC].

4.3.
A Plücker Embedding for the Symplectic Grassmannian.Consider the irreducible fundamental Sp 2n -module V ω k of highest weight ω k .Let ν ω k denote a highest weight vector in V ω k .We have the standard representation V ω 1 ≃ C 2n and the canonical embedding, Notice that unlike in the case of type A, the image of V ω k under the above embedding is not the whole of k C 2n .This also implies that the projectivization P(V ω k ) of V ω k does not coincide with the whole projective space P k C 2n .Nonetheless, we choose an embedding of SpGr(k, 2n) into the latter space as described below.This is because we want to keep all the Plücker coordinates in the defining relations for easy comparison with the type A case.
Let U k denote an element in Gr(k, 2n) such that U k = span(u 1 , . . ., u k ), for some u 1 , . . ., u k ∈ C 2n .Consider the Plücker embedding, The Plücker embedding of SpGr(k, 2n) that we are considering is the composition of the above embeddings.It turns out that the isotropic condition on the elements U k ∈ SpGr(k, 2n) leads to linear relations among the 2n k Plücker coordinates X J on Gr(k, 2n).This follows from the work of De Concini [DEC], and it implies that these linear relations cut out the image of SpGr(k, 2n) from Gr(k, 2n) as discussed in the following two subsections.4.4.Reverse-admissible Minors and their Correspondence with the Symplectic PBW Tableaux.Following [DEC], we consider now the variety V whose points over C are the m-tuples (v 1 , . . ., v m ) of vectors in W such that v i , v j = 0 for all 1 ≤ i, j ≤ m, where , is the symplectic form defined above.The variety V is therefore equivalently the variety of 2n × m matrices M with coefficients in C such that M T ΨM = 0, where Ψ is the matrix of the form , and M T denotes the transpose of the matrix M.
We recall the following result: DEC]).In the ring A, the coordinate ring of the variety V, any minor can be expressed as a linear combination of admissible minors of the same size and involving the same columns.
To find a connection of the variety V to SpF 2n , the complete symplectic flag variety, we recall a few more results from [DEC].The isotropic Stiefel variety W m,n is the open set in V whose points over C are the m-tuples of vectors (v 1 , . . ., v m ) in W such that (v 1 , . . ., v m ) span an isotropic subspace of W of dimension equal to min(n, m).

Corollary 4.3 ([DEC]
).Let A ′ be the ring of global polynomial functions on W m,n , then A ′ = A, where A is the coordinate ring of V.
Also there is a natural morphism g : W n,n → SpF 2n given by where U (v 1 ,...,vt) = {linear span of v 1 , . . ., v t } for some t vectors v 1 , . . ., v t in W.

Proposition 4.4 ([DEC]
).The morphism g : W n,n → SpF 2n is a principal B bundle, where B is the Borel subgroup of upper triangular elements in Gl(n).
Proposition 4.4 implies that we actually have SpF 2n = W n,n /B.This and Corollary 4.3 imply that C[SpF 2n ] is a sub-ring of A, i.e., it is the ring of invariants in A under the group action of B on W n,n .Right canonical minors are those with i's on the i-th column i.e., minors of the form (i k , . . ., i 1 |1, . . ., k).These are all we need to work with in C[SpF 2n ] (see [DEC,Theorem 4.8]).We will therefore restrict to these minors, in that we will write (i k , . . ., i 1 ) instead of (i k , . . ., i 1 |1, . . ., k) and (I 2 , I 1 ) instead of (I 2 , I 1 |1, . . ., k).We will also write 'minor' instead of 'right canonical minor' for brevity.Now we would like to find a connection of the admissible minors to the symplectic PBW tableaux.For this, we choose an equivalent but different version of these minors, which we call reverse-admissible.In this regard, keeping the same notation as above, we would like to give the following definition.
It turns out that Proposition 4.1 also holds for reverse-admissible minors: Proposition 4.6.In the ring C[SpF 2n ], any minor can be expressed as a linear combination of reverse-admissible minors of the same size and involving the same columns.
To prove this proposition, we first recall Proposition 1.8 of [DEC], and a modified version of Definition 1.4 of [DEC] which gives a total ordering on the set of right canonical minors.
From a k ×k minor (I 2 , I 1 ) (which is not necessarily reverse-admissible), we describe how to obtain a single column length k PBW tableau (which is not necessarily symplectic).For this, we first compute the minor (I 2 , I 1 ) according to Equation (4.3).We then put every entry which is less than or equal to k at its position in the column of length k, and every other entry should be put in such a way that it is bigger than entries below it.For example, the single column length 4 PBW tableaux corresponding to the 4 × 4 computed minors (2, 2, 1, 1), (3, 3, 1, 1), and (4, 4, 1, 1) are respectively the tableaux: Moreover, we can also recover a k × k minor (I 2 , I 1 ) from the corresponding single column length k PBW tableau.To do this, we put every element i for which i belongs to our tableau in I 2 , and all other elements which appear in the tableau without bars are put in I 1 .We prove the following result.
Proposition 4.10.The reverse-admissible k × k minors are in a weight preserving bijection with the single column length k symplectic PBW tableaux.
Proof.We first show that the tableaux corresponding to the reverse-admissible minors (I 2 , I 1 ) satisfy the conditions of Definition 3.1.Conditions (i) and (ii) are clearly always satisfied up to a reordering of the entries appearing in the computed minor of (I 2 , I 1 ) as described above.It remains to verify condition (iii) of Definition 3.1, namely, we want to show that whenever we have a pair (i, i) with i < k in the computed minor, then after re-ordering to satisfy (i) and (ii), the position of i is above that of i.
For the other direction, assume we are given a single column symplectic PBW tableau.For all i in the tableau, put i in I 2 , and put the rest of the indices in I 1 .Also, for all (i 1 , . . ., i λ ) for which we have (i 1 , . . ., i λ ) in the column, let j t be the position of i t for all 1 ≤ t ≤ λ.The tableau being a symplectic PBW tableau implies j t ≤ i t for all 1 ≤ t ≤ λ.Also, we note that j t ∈ {1, . . ., n}\(I 1 ∪ I 2 ) for all 1 ≤ t ≤ λ and hence the set {j 1 , . . ., j λ } is the minimal set with the required properties.Hence (I 2 , I 1 ) is reverse-admissible.This gives the bijection.The fact that this bijection is weight preserving is straight forward.Proposition 4.10 allows us to use the notions reverse-admissible minors and single column symplectic PBW tableaux interchangeably.We do this henceforth.(4.7) The composition of the above embeddings is the Plücker embedding of SpF 2n that we are considering.
Consider the polynomial ring C[X j 1 ,...,j d ] generated by all the elements X j 1 ,...,j d , d = 1, . . ., n and 1 We want to describe the defining (or vanishing) ideal in C[X j 1 ,...,j d ] for SpF 2n under the composition of the above embeddings.We first recall how this ideal relates to that of the type A flag variety in the following remark.
Remark 4.11.Consider g = sl 2n (C) and − 1 be a sequence of increasing numbers.For an sl 2n dominant integral weight λ = 2n−1 s=1 m s ω ds , we consider the corresponding partial flag variety, which we denote by SLF λ .This variety is known to coincide with the projective variety of partial flags where U k ⊂ C 2n for 1 ≤ k ≤ s.We also consider the Plücker embedding of this variety.
Remark 4.13.In both Expressions (4.8) and (4.9), the following equality is assumed: Let I denote the ideal of the polynomial ring C[X j 1 ,...,j d ] generated by the Plücker relations R t L,J and the symplectic relations S (I 2 ,I 1 ) .We have: Theorem 4.14 ( [DEC]).The ideal I is the defining ideal of SpF 2n with respect to the Plücker embedding, SpF 2n ֒→ n k=1 P k C 2n .It is a prime ideal.
Proof.By Remark 4.11, we have that for any λ of the form λ = n s=1 m s ω s , with all m s = 0, the inclusion I(SLF λ ) ⊂ I(SpF 2n ) of defining ideals holds.It therefore follows that the relations R t L,J are satisfied on SpF 2n , since they are satisfied on the type A 2n−1 partial flag variety SLF λ according to [FUL,Lemma 1,p. 132].The relations S (I 2 ,I 1 ) come from Equation (4.5) from the proof of Proposition 4.6, so they are clearly satisfied.
It follows from [DEC,Theorem 4.8] that the relations R t L,J and S (I 2 ,I 1 ) are enough to express every non standard symplectic tableau as a linear combination of symplectic standard tableaux in the homogeneous coordinate ring of SpF 2n .That is to say, these relations provide a straightening law for this coordinate ring.This therefore implies that these relations generate the defining ideal of SpF 2n .This ideal is prime since SpF 2n is irreducible.
Example 4.15.For SpF 4 , the ideal I is generated by the following relations: 4.6.A Basis for the Coordinate Ring.Let C[SpF 2n ] denote the multi-homogeneous coordinate ring of SpF 2n .One has the following statement: where the multiplication V * λ ⊗ V * µ → V * λ+µ is induced by the injective homomorphism of modules: λ is given by the classical Borel-Weil theorem.
In order to prove this proposition, we first prove a useful result about the PBW-degree introduced in Section 2. This will also be important for the constructions in Section 5.
The following is the central definition in this light: Definition 4.17.For any sequence J = (j 1 < • • • < j k ) ⊂ {1, . . ., n, n, . . ., 1}, 1 ≤ k ≤ n, the PBW-degree of J is given by the formula: deg J = #{r : j r > k}. (4.11) The PBW-degree of the minor (I 2 , I 1 ) is the PBW-degree of its computed minor as given in Equation (4.3).The PBW-degree of an element X J is defined to be the PBW-degree of J.The PBW-degree of an element X T from (4.10) above is the sum of the PBW-degrees of the elements X T 1,j ,...,T µ j ,j appearing in the product.
Remark 4.18.We would like to give an interpretation of the above definition of PBW-degree in terms of the definition given in Section 2. is given by the elements w J := w j 1 ∧ • • • ∧ w j k labelled by all sequences J = (j 1 < • • • < j k ) ⊂ {1, . . ., n, n, . . ., 1}.Here the highest weight vector ν ω k is identified with In fact, p can be chosen as f i 1 ,j 1 • • • f is,js for some root vectors f it,jt , 1 ≤ t ≤ s.This degree therefore corresponds to the number of indices in J that are greater than k.The component (L ω k ) s is spanned by the elements w J with deg w J ≤ s.Therefore, the images of the elements w J with deg w J = s give a basis of (L ω k ) s /(L ω k ) s−1 ֒− → L a ω k , the PBW degenerate module.Let w a J denote these images and let X a J denote the dual elements.Now we consider the irreducible fundamental sp 2n -module V ω k ⊂ L ω k ≃ k C 2n .Consider an arbitrary element in V ω k and write it as a sum of pure wedge tensors.Then find the minimal sl 2ncomponent which contains all the summands.We consider the maximal sl 2n -degree of each of these summands.This is the PBW-degree induced on sp 2n .By the PBW theorem, this induced PBW-degree is compatible with the one on sl 2n ( [BFK]).
Remark 4.19.We can obtain the PBW-degree of (I 2 , I 1 ) directly from the subsequences I 1 and I 2 without first computing the minor.For this, we use the formula: (4.12) To see that the degrees given in Equations (4.11) and (4.12) agree, we only need to consider the PBW-degree of the computed minor L of (I 2 , I 1 ).Indeed from Equation (4.3), we have that: = deg(I 2 , I 1 ).
We prove the following fundamental lemma.
Lemma 4.20.Following the notation of Definition 4.12, the PBW-degree of each of the summands appearing in the right hand side of is greater than or equal to the PBW-degree of the term X (I 2 ,I 1 ) on the left hand side, whenever (I 2 , I 1 ) is not reverse-admissible.
We end this section by proving Proposition 4.16.
Proof of Proposition 4.16.From Theorem 3.16, we have that #{T : T ∈ SyST λ } = dim V λ , so it remains to show that the elements X T , T ∈ SyST λ , span C[SpF 2n ] λ .Suppose we are given an element X T with T / ∈ SyST λ .We claim that X T can be written as a linear combination of elements X T ′ with T ′ ∈ SyST λ .
From Propositions 4.6 and 4.10, it suffices to consider only PBW tableaux T that are symplectic but not PBW-semistandard.Recall that the condition of PBW-semistandardness is defined between neighbouring columns of T. It is therefore sufficient to consider any two such columns of T which violate this condition.Let these columns be denoted by L and J, with length of L equal to p and length of J equal to q such that p ≥ q.We first of all apply the Plücker relation (4.8), to express the product X L X J as a sum of products X L (i) X J (i) , that is: However, after exchanging, it may happen that for one of the variables X L (i) or X J (i) , the corresponding L (i) or J (i) is no longer a single column symplectic PBW tableau, that is to say, the corresponding minor (I 2 , I 1 ) (i) is not reverse-admissible (by Proposition 4.10).In this case, we apply the symplectic relation (4.9), to replace such a variable with a sum of variables corresponding to reverse-admissible minors.
Now from Lemma 4.20 and from the proof of Proposition 4.12 of [FEI], we see that Therefore in C[SpF 2n ] λ , any X T with T / ∈ SyST λ , can be written as a linear combination of X T ′ with the sum of PBW-degrees of X T ′ bigger than or equal to that of X T .This implies the claim since the sum of PBW-degrees of fixed shaped tableaux is bounded from above.
5. The Complete Symplectic PBW Degenerate Flag Variety; Symplectic Degenerate Relations and a Basis for the Coordinate Ring In this section, we describe the complete symplectic PBW degenerate flag variety following [FFIL].We then prove results on a basis for the multi-homogeneous coordinate ring and the generating set of relations for the defining ideal with respect to the Plücker embedding.
5.1.PBW Degenerate Flag Varieties; a Brief Description.Let G a be a Lie group corresponding to the PBW degenerate Lie algebra g a .Let us briefly describe the Lie group G a .Let G a be the additive group of the field C and let M = dim n − .The Lie group G a is a semidirect product G M a ⋊ B of the normal subgroup G M a and the Borel subgroup B. For any dominant, integral weight λ, there exist induced g a -and G a -module structures on V a λ .The group G a therefore acts on P(V a λ ), the projectivization of V a λ .The PBW degenerate flag variety ( [FEI]) is defined to be the closure of the G a -orbit through the highest weight line, that is to say: In particular, for λ = ω k , we have the case of Grassmann varieties.For g = sl n+1 (C) and . This is true because all the fundamental weights ω k are co-minuscule in type A, and hence the radical corresponding to each ω k is abelian.So, the PBW degenerate flag variety in type A is embedded into the product of Grassmannians ( [FEI,Proposition 3.3]).
For g = sp 2n (C) and G = Sp 2n (C), we denote by SpF ω k and SpF a ω k the symplectic original and PBW degenerate flag varieties.It is known that except for the weight ω n , all the other fundamental weights of sp 2n are not co-minuscule.This implies that the radicals corresponding to the weights ω 1 , . . ., ω n−1 are not abelian.So, the varieties SpF ω k and SpF a ω k are not isomorphic in general, except for the case k = n.
The variety SpF a ω k is the symplectic PBW degenerate Grassmannian, which we will henceforth denote by SpGr a (k, 2n), to match the notation we have used for the original symplectic Grassmannian SpGr(k, 2n).For a dominant, integral and regular weight λ, the complete symplectic PBW degenerate flag variety is the variety: . Recall that this is the PBW degeneration of the projective variety SpF 2n from the previous section.For the rest of the paper, we set our focus on the variety SpF a 2n .
5.2.The Complete Symplectic PBW Degenerate Flag Variety.The material stated in this subsection is based on [FFIL].Recall the vector space W with the fixed basis {w 1 , . . ., w 2n }.To begin with, let us recall an important result about the linear algebra realization of the symplectic PBW degenerate Grassmannian SpGr a (k, 2n).
Remark 5.7.Observe that the degenerate relations R t;a L,J and S a (I 2 ,I 1 ) are homogeneous with respect to the PBW-degree.This follows directly from Definition 5.3 since the terms in R t;a L,J and S a (I 2 ,I 1 ) are those of minimal PBW-degrees picked from the relations R t L,J and S (I 2 ,I 1 ) respectively.
where the multiplication (V a λ ) * ⊗ (V a µ ) * → (V a λ+µ ) * for any two dominant integral weights λ and µ in the algebra on the right hand side is induced by the embedding of g a -modules, V a λ+µ ֒→ V a λ ⊗ V a µ (according to Lemma 2.6).Recall that this embedding is compatible with the PBW-degree and hence the algebra λ∈P + (V a λ ) * is PBW-graded.The isomorphism of PBW-graded algebras follows from [FFIL,Theorem 1.2,Theorem 1.4], which respectively give the flatness of the degeneration and an analogue of the Borel-Weil theorem for the PBW degenerate module V a λ .
We have the elements X a j 1 ,...,j k ∈ ( k C 2n ) * (Remark 4.18).By restricting to V a ω k ⊂ k C 2n , we can consider these elements also in (V a ω k ) * .
Proposition 5.8.The degenerate relations R t;a L,J and S a (I 2 ,I 1 ) are both zero in the coordinate ring C[SpF a 2n ] with respect to the composition of the embeddings in (5.1).
Proof.We know that the elements X a j 1 ,...,j k ∈ ( k C 2n ) * satisfy relations R t;a L,J in C[SpF a 2n ] with respect to (5.1) (see [FEI,Lemma 4.1,Theorem 4.10]).From Theorem 4.14, we have that the elements X j 1 ,...,j k satisfy relations S (I 2 ,I 1 ) in C[SpF 2n ] with respect to the embeddings in (4.7).
The coordinate ring C[SpF a 2n ] is PBW-graded and SpF a 2n is a flat degeneration of SpF 2n (again [FFIL,Theorem 1.2]), hence the lowest PBW-degree term relations from S (I 2 ,I 1 ) are equal to 0. By definition, these lowest term relations are the relations S a (I 2 ,I 1 ) .Recall SyST λ , the set of all symplectic PBW-semistandard tableaux of shape λ = (λ 1 ≥ • • • ≥ λ n ≥ 0).To each T ∈ SyST λ , we associate the monomial element X a T 1,j ,...,T µ j ,j ∈ (V a λ ) * , and call such an element, the symplectic PBW-semistandard monomial.We prove the following result.
Theorem 5.9.The elements X a T , T ∈ SyST λ , form a basis of C[SpF a 2n ] λ .Proof.From Theorem 2.5 and Theorem 3.16, we know that dim V a λ = #{T : T ∈ SyST λ }.It therefore remains to prove that the elements X a T , T ∈ SyST λ span C[SpF a 2n ] λ .From Proposition 5.8, we know that the elements X a j 1 ,...,j d satisfy relations R t;a L,J and S a (I 2 ,I 1 ) in C[SpF a 2n ].We are going to use these relations to write an element X a T for which T is not symplectic PBW-semistandard as a linear combination of elements X a T ′ with T ′ symplectic PBW-semistandard.For this, we first follow [FEI] to define an order on the set of PBW tableaux of shape λ.Say that T (1) > T (2) if there exist i 0 , j 0 such that T (1) i 0 ,j 0 > T (2) i 0 ,j 0 and T (1) i,j = T (2) i,j if (j > j 0 , i = i 0 ) or (j = j 0 , i > i 0 ).
Since the condition of PBW-semistandardness is defined between every two adjacent columns, we can reduce the proof to any two such columns.Supposing we are given two such columns L and J that form a PBW tableau that is not PBW-semistandard.We are going to use the degenerate Plücker relations R t;a L,J to obtain terms corresponding to smaller PBW tableaux.In fact, let L = (l 1 , . . ., l p ) and J = (j 1 , . . ., j q ) with p ≥ q.From the proof of Proposition 4.12 of [FEI], we have that the term X a l 1 ,...,lp X a j 1 ,...,jq is present in the relation R t;a (l 1 ,...,lp),(j 1 ,...,jq) and that all the other terms correspond to smaller PBW tableaux with respect to the order " > " on the set of PBW tableaux.The only thing that remains is to show that we can write a term corresponding to a PBWsemistandard tableau that is not symplectic as a linear combination of terms corresponding to symplectic PBW-semistandard tableaux.For this, we use the symplectic degenerate relations.Indeed, let L ′ be any non symplectic column that appears as a result of the exchange process during the application of the degenerate Plücker relations above.Then from Lemma 4.20, the term X L ′ is among the terms with minimal PBW-degree in S L ′ .This means that the term X a L ′ is present in the relation S a L ′ since this relation is obtained by picking up terms of minimal PBW-degree from S L ′ .We can therefore use the relation S a L ′ to replace terms corresponding to non symplectic columns.We claim that the new columns that arise in this process are smaller than L ′ with respect to the order " > ".For this, recall the definition of the relations S L ′ and S a L ′ and the used notation.
From Proposition 4.6 and Lemma 4.20, we know that the largest element which can be removed from L ′ = (I 2 , I 1 ) ′ is γ b and that γ b < k.Hence we also know that the PBW-degree goes up only when there exists i with h 0 ≤ i ≤ b such that γ ′ i > k, among the new entries.Therefore since we are using relations S a L ′ , it suffices to consider the case γ ′ b ≤ k.For any given term X a L ′′ in S a L ′ apart from X a L ′ , and for the corresponding sequence L ′′ , let f be the position of γ ′ b after rearranging the entries to form a single column symplectic PBW tableau.Clearly, we need to begin comparing the entries of the columns L ′ and L ′′ starting from position f downwards.To see this, recall that since γ ′ b ≤ k, then f = γ ′ b .This implies that the entry at position f in L ′ is different from f since γ ′ b ∈ {1, . . ., n} \ (I 1 ∪ I 2 ) ′ with L ′ = (I 2 , I 1 ) ′ .Let L ′ f denote the entry at position f in single column symplectic PBW tableau L ′ .We have L ′ f > f = γ ′ b .Moreover, all entries below position f (if any), are pairwise equal in L ′ and L ′′ .This implies that L ′ > L ′′ , and hence the claim is proved.5.5.Defining Ideal.Let I a ⊂ C[X a j 1 ,...,j d ] be the ideal generated by the degenerate relations R t;a L,J and S a (I 2 ,I 1 ) .The following is the main statement of this paper.Theorem 5.10.The ideal I a is the defining ideal of the variety SpF a 2n under the Plücker embedding, SpF a 2n ֒− → n k=1 P k C 2n .
Proof.From Theorem 5.9, we see that the relations R t;a L,J and S a (I 2 ,I 1 ) in I a are enough to express every monomial in Plücker coordinates as a linear combination of symplectic PBW-semistandard monomials (i.e., these relations provide a straightening law for C[SpF a 2n ]).Following the idea of the proof of Theorem 7 in [CFFFR], this implies that the ideal I a is the defining ideal of SpF a 2n since otherwise, it would imply that the symplectic PBW-semistandard monomials are not a basis for C[SpF Corollary 5.12.The ideal I a is a prime ideal of the polynomial ring C[X a j 1 ,...,j d ].Proof.This follows directly from Theorem 5.10 and the fact that the variety SpF a 2n is irreducible (see [FFIL,Corollary 5.6]).
Remark 5.13.We know from [FFIL,Theorem 1.2] that SpF a 2n is a flat degeneration of SpF 2n .We would like to give a formulation of this result in terms of the results of this paper.Let s be a variable.We follow [FEI] to define an algebra Q s over the ring C[s] as a quotient of the polynomial ring C[s][X a j 1 ,...,j d ], d = 1, . . ., n by the ideal I s generated by quadratic relations R t;s L,J and linear relations S s (I 2 ,I 1 ) which are s-deformations of the relations R t L,J and S (I 2 ,I 1 ) .Let R t L,J = i X L (i) X J (i) and S (I 2 ,I 1 ) = i X (I 2 ,I 1 ) (i) , then: R t;s L,J = s − min i (deg L (i) +deg J (i) ) i s deg L (i) +deg J (i) X L (i) X J (i) , S s (I 2 ,I 1 ) = s − min i (deg (I 2 ,I 1 ) (i) ) i s deg (I 2 ,I 1 ) (i) X (I 2 ,I 1 ) (i) .
We have Q s /(s) ≃ C[SpF a 2n ], and Q s /(s − u) ≃ C[SpF 2n ] for u = 0.Moreover, following Theorem 5.9, one checks that the elements X T , T ∈ SyST λ , λ ∈ P + , form a C[s] basis of Q s , hence showing that Q s is free over C[s].

k
) * be the Plücker coordinate labelled by J.These Plücker coordinates are k × k minors of the 2n × k matrix representing the subspace U k .The image of Gr(k, 2n) under the above embedding is identified with the set of all Plücker vectors in P ( 2n k )−1 , which are vectors whose coordinates are the 2n Plücker coordinates X J .Now we consider the embedding of SpGr(k, 2n) into the Grassmannian Gr(k, 2n), i.e., SpGr(k, 2n) ֒→ Gr(k, 2n) ֒→ P k C 2n .

5. 4 .
A Basis for the Coordinate Ring.Let C[SpF a 2n ] denote the multi-homogeneous coordinate ring of the complete symplectic PBW degenerate flag variety.Then one has C[SpF a 2n a 2n ].Remark 5.11.From Theorem 5.10, we can now write down the multi-homogeneous coordinate ring of SpF a 2n as a quotient of the polynomial ring C[X a j 1 ,...,j d ] by the ideal I a , i.e., C[SpF a 2n ] = C[X a j 1 ,...,j d ]/I a ≃ λ∈P + (V a λ ) * .
Consider g = sl 2n (C) with Cartan decomposition sl 2n = n + sl 2n ⊕ h sl 2n ⊕ n − sl 2n .Consider the irreducible fundamental sl 2n -module L ω k of highest weight ω k .We identify L ω k with k C 2n .Recall the standard basis {w 1 , . . ., w 2n } of C 2n .Then a basis of