Diamond representations of $\mathfrak{sl}(n)$

In \cite{W}, there is a graphic description of any irreducible, finite dimensional $\mathfrak{sl}(3)$ module. This construction, called diamond representation is very simple and can be easily extended to the space of irreducible finite dimensional ${\mathcal U}\_q(\mathfrak{sl}(3))$-modules. In the present work, we generalize this construction to $\mathfrak{sl}(n)$. We show this is in fact a description of the reduced shape algebra, a quotient of the shape algebra of $\mathfrak{sl}(n)$. The basis used in \cite{W} is thus naturally parametrized with the so called quasi standard Young tableaux. To compute the matrix coefficients of the representation in this basis, it is possible to use Groebner basis for the ideal of reduced Pl\"{u}cker relations defining the reduced shape algebra.


Introduction
In this paper, we consider the irreducible finite dimensional representations of the Lie algebra sl(n) = sl(n, C). Of course these representations are well known and there are very explicit descriptions for them, for instance in [FH].
Date: 07/11/05. The direct sum S • of all the simple modules has a natural realization as the shape algebra of sl(n), i.e. as the algebra C[SL(n)] N + of polynomial functions on the group SL(n), which are invariant under the right multiplication by upper triangular matrices. Let g be an element in SL(n), denote δ (s) i 1 ,...,is (g) the determinant of the submatrix of g obtained by considering the s first columns of g and the rows i 1 < · · · < i s , then S • is generated as an algebra by the functions δ (s) i 1 ,...,is . More precisely, it is the quotient of C[δ (s) i 1 ,...,is ] by the ideal P (δ) generated by the Plücker relations.
Generally a parametrization of a basis for S λ is given by the set of semi-standard Young tableaux T of shape λ i.e. with a n−1 columns of size n − 1, . . . , a 1 columns of size 1.
Using this description, we give here a natural ordering on the set of variables δ (s) i 1 ,...,is , we determine the Groebner basis of P (δ) for this ordering, getting the corresponding basis of the quotient as monomials δ T , for T semi-standard.
Thus the action of upper triangular matrices on this basis can be easily computed. (See for instance the description given in [LT]).
On the other hand, in [W], N. Wildberger gave a really different presentation of the simple sl(3)-modules. This description is based on the construction of the diamond cone for sl(3), it is an infinite dimensional indecomposable module for the Heisenberg Lie algebra with a very explicit basis. The matrix coefficients are integral numbers and fixing the highest weight λ, it is easy to build the corresponding representation of sl(3), on the submodule generated by this vector in the diamond cone.
In this paper, we extend this presentation to sl(n). In fact the diamond cone module is a quotient of the shape algebra. We call this quotient the reduced shape algebra. It is the quotient of C[δ (s) i 1 ,...,is ] by the ideal P red (δ) sum of the ideal of Plücker relations and the ideal generated by δ (s) 1,...,s − 1.
With the same approach as above, we define a new ordering on the variables δ (s) i 1 ,...,is , with this ordering, we can compute the Groebner basis for P red (δ) and the corresponding basis for the quotient : the set of monomials δ T , for some Young tableaux T called here quasi-standard.
The action of the upper triangular matrices on this basis is easy to compute : this gives us the diamond cone for sl(n).
In order to refind the complete sl(n)-modules, we have to define a symmetry on each S λ and on the corresponding submodule in the reduced shape algebra. This symmetry exchanges the role of N + and N − and we get the complete sl(n) representation.
Unfortunately, this symmetry corresponds to a modification of the ordering on Young tableaux, thus, if n > 3 to a different basis in S λ . The n − action on the first base is not so simple as in [W].

Usual (algebraic) presentation of the sl(n) simple modules
Let us consider the Lie algebra sl(n) = sl(n, C): it is the set of n × n traceless matrices, it is the Lie algebra of the Lie group SL(n) of n × n matrices, with determinant 1. The Cartan algebra h is the space of diagonal matrices: θ n   , θ j ∈ C, θ 1 + · · · + θ n = 0    .
We put α i (H) = θ i . The root system of sl(n) is the set of linear form on h generated by the α i − α j , (i = j).
The usual basis ∆ for the root system is given by : The root space corresponding to the simple root η i = α i − α i+1 is generated by the upper triangular matrix: The root space corresponding to −η is generated by lower triangular matrix: these matrices generate sl(n) as a Lie algebra.
The set of simple sl(n)-modules up to equivalence is isomorphic to the set of dominant integral weights. More precisely, sl(n) acts naturally on V = C n (with canonical basis e 1 , . . . , e n ), thus also on the totally antisymmetric tensor products ∧ j V (j = 1, . . . , n − 1) and on the symmetric tensor products Sym a j (∧ j V ) and finally on Sym a 1 (V ) ⊗ Sym a 2 (∧ 2 V ) ⊗ · · · ⊗ Sym a n−1 (∧ n−1 V ).
With this construction, we get each simple sl(n)-module, and two distinct weights λ, λ ′ give rise to inequivalent simple sl(n)-modules. This action gives rise by exponentiation to a representation of SL(n). Let us put where ε n = 1 if n 2 is even and ε n = e iπ n if n 2 is odd. Then Ω belongs to SL(n). In fact, this matrix, acting by adjoint action generates the longest element of the Weyl group of SL(n). It correspnds to a change in the choice of simple roots and nilpotent subalgebras n + and n − , if X = [x ij ] is a strictly upper triangular matrix, Ω −1 XΩ = x (n+1−i)(n+1−j) is strictly lower triangular. Let us put: v λ − = (e n ) a 1 ⊗ (e n ∧ e n−1 ) a 2 ⊗ · · · ⊗ (e n ∧ · · · ∧ e 2 ) a n−1 = ε −|λ| n Ω.v λ , with |λ| = a 1 + 2a 2 + · · · + (n − 1)a n−1 . Then v λ − is a lowest weight vector in S λ (V ).

The shape algebra: abstract algebraic presentation
Let us put: Since we have an explicit realization of each highest weight vector, it is possible to define a natural comultiplication ∆ on S • (V ), just by defining as the unique sl(n)-morphism sending v λ on ⊗ · · · ⊗ (e 1 ∧ · · · ∧ e n−1 ) b n−1 +c n−1 .
Since each isotypic component of the SL(n) module S • (V ) is simple the multiplication m is characterized by this relation and the condition We shall call shape algebra of SL(n) the algebra S • (V ) equipped with the above multiplication.
This theorem is well known. There is a complete proof in [FH] p. 235, this result is cited by Towber in [LT] as a theorem due to Kostant.
We define a symmetry τ in S • just by putting: Since the multiplication is a morphism of sl(n) and SL(n) module, τ (vv ′ ) = τ (v)τ (v ′ ). Especially, we can define the multiplication just as above by fixing . Now for each matrix A in sl(n), ΩAΩ = τ A is the matrix defined by a central symmetry on the entries of A:

The shape algebra: geometric presentation
The shape algebra can also be viewed as an algebra of functions on a quotient SL(n)/N + of the Lie group SL(n). Denote N + the subgroup Let us consider the space C[SL(n)] = C[g ij ]/(det − 1) of all polynomial functions f with respect to the entries g ij of the matrix g ∈ SL(n). There is a SL(n) × SL(n) action on this space, defined as follows: Since this space is generated by the invariant finite dimensional subspaces of class of functions with degree less than N (N = 0, 1 . . . ), this action is completely reducible in a sum of finite dimensional simple SL(n) × SL(n) modules. The highest vector for these modules are class of functions f such that: But, let us consider the restriction of f to the dense set of g such that, for s = 1, . . . , n, δ 1,2,...,s (g) = 0. On this set, using the Gauss method, we can reduce g to a diagonal matrix, getting: The highest weight is (λ, λ) with λ = a i ω i , that means f is a polynomial function in the variables 1,...,n−2 (g) , 1 δ (n−1) 1,2,...,n−1 (g) , homogeneous with degree a 1 + · · · + a n−1 , a 2 + · · · + a n−1 , . . . , a n−1 , 0, i.e. the function f is a multiple of the function: Acting with only the first factor SL(n) on these functions, we get all the polynomial N + right invariant functions on SL(n). Due to the form of the bi-invariant functions f , these functions are polynomial functions in the δ-variables : ..,is ]/P (δ), where P (δ) is an ideal. Moreover each irreducible representation of SL(n) happens exactly one times in this space, thus as a vector space, Acting on δ λ (λ = i a i ω i ) on the left by N − = t (N + ), we get polynomial functions which contains only monomials of the form: Let us call V a 1 ,...,a n−1 the space of such functions. In view of our description, it is a simple module and the isotypic component of type λ in C[SL(n)] N + .
The shape algebra is isomorphic to the algebra O(SL(n)/N + ) of the regular functions on the homogeneous space SL(n)/N + .
The ideal P (δ) is the ideal generated by the Plücker relations written on the δ functions.

Remark 1.
In this presentation of S • (C n ), the SL(n) action on the elements of the shape algebra, viewed as a polynomial function f is very natural since it is just: The symmetry τ can be directly implemented in the space C[SL(n)] N + . Indeed τ is up to conjugation by Ω, a morphism of SL(n) modules and the formula τ (e 1 ∧ · · · ∧ e s ) = ε s n e n ∧ · · · ∧ e n+1−s becomes here τ (δ (s) 1,2,...,s ) = ε s n δ (s) n,(n−1),...,(n+1−s) . But, if we put for any regular function f on SL(n), (θf )(g) = f (Ωg), we define a bijection from C[SL(n)] N + into itself such that

The shape algebra : Combinatorial presentation
The usual basis of S λ (V ) are parameterized by the semi standard Young tableaux with shape λ. Let us be more precize: We can naturally associate to each δ variable a column C: Then if we identify two Young tableaux which differ only by a permutation of their columns, the set of Young tableaux defines a linear basis for the algebra C[δ (s) i 1 ,...,is ]:

. . .
(p 1 ≤ p 2 ≤ · · · ≤ p k ). That means, we read the Young tableau from right to left, using the following convention: if two different columns C and C ′ have the same height, we put in the first place in T the column , and i r < i ′ r . The Plücker relations are quadratic in the δ variables, they correspond to linear combination of Young tableaux with two columns, for instance, we get for sl(3) the following relation between tableaux: In order to describe a basis for the quotient space: we will use the notion of Groebner basis [CLO].
Let us consider the algebra C[X 1 , . . . , X k ] of polynomials in the variable X i and an ideal I of C[X 1 , . . . , X k ].
Suppose we fix an ordering on the set of monomials X α 1 1 . . . X α k k in C[X 1 , . . . , X k ] (for instance by using the lexicographie ordering on words α 1 . . . α k if we put X 1 < X 2 < · · · < X k ). Then any polynomial g has an unique leading term LT (g); the greatest monomial happening in g for this ordering.

Definition 1.
A finite subset {g 1 , . . . , g k } of an ideal I is said to be a reduced Groebner basis for I if and only if the leading term of any element of I is divisible by one of the leading term of g i and if for all g i no monomial of g i is divisible by the leading term of some g j j = i.
If {g 1 , . . . , g k } is a reduced Groebner basis for I, then the set of (classes of) monomials which are not divisible by any monomials LT (g i ) (i = 1, . . . , k) is a basis of the quotient C[X 1 , . . . , X k ]/I. Following [FH], we know there is in the ideal P (δ) the following elements for any p ≥ q ≥ r: j 1 ,...,jp if i p = j p , . . . , i r+1 = j r+1 and i r < j r . We put the lexicographic ordering on the monomials δ T in C[δ (p) i 1 ,...,ip ].

Remark 2.
In [FH], an ordering < on Young tableaux is defined,in fact our ordering is the reverse ordering since: Recall that a Young tableau is semi standard if its entries are increasing along each row (and strictly increasing along each column). It is well known that the set of semi standard Young tableau gives a basis for C[δ (p) i 1 ,...,ip ]/P (δ) (see [FH] for instance).
Our ordering defines an unique Groebner basis for P (δ). We shall now build this basis.
For any non semi standard Young tableau T with 2 columns, there exists an element in P (δ) of the form ( * ). This relation can be written as: Each T j has the same shape as T but some of them can be non semi standard. We repeat the construction for each non semi standard T j and finally we get, for each non semi standard T with 2 columns, an element f T in P (δ) such that the leading term of f T is δ T and all the monomials of f T have the form a.δ T ′ with T ′ semi standard and δ T ′ < δ T .

Theorem 2.
The set G = {f S , S non semi standard with 2 columns} is the reduced Groebner basis of P (δ) for our ordering.

Proof:
First denote NS the set of all monomials δ T with T non semi standard. Since each non semi standard T has 2 consecutive columns such that the sub tableau defined by these 2 columns is non semi standard, δ T is divisible by one of the δ S , i.e. by one of the leading term of f S .
Thus the ideal < δ S > generated by the leading terms of G contains the vector space span(NS).
Conversely let T be a semi standard Young tableau. Suppose T belongs to the ideal < LT (P (δ)) > generated by the leading terms of all the f in P (δ). That means: If any T ′ is semi standard we keep this relation. If some of the T ′ are non semi standard, then δ T ′ is in < δ S > thus in < LT (P (δ)) > and we repeat the construction for δ T ′ . We get finally: This implies that But this is impossible, since the set {δ T , T semi standard} is a basis for C[δ (p) i 1 ,...,ip ]/P (δ) Thus: Moreover, since any monomial in f S is either δ S or a T δ T with T semi standard, it can not be divisible by a δ S ′ with S ′ = S, S ′ non semi standard with two columns. This proves our theorem.
The usual basis of the shape algebra S • (V ) by semi standard Young tableaux can thus be described as a natural basis of a the quotient of the polynomial algebra C[δ Especially, we can write the action of any element of the Lie algebra sl(n) on any polynomial function with variables δ (s) i 1 ,...,ip ] as the derivation: Finally, the Cartan algebra acts on f as the derivation This action defines the action on the quotient by P (δ), since we have a Groebner basis for the ideal P (δ), the quotient action on the basis of semi standard Young tableaux reduces to compute the canonical form of the polynomial X α f or Hf , this is easy to do with usual computer software.
As an illustration, we gives a graphic description of the N + part of the adjoint representation S ω 1 +ω 2 (C 3 ) of sl(3) (see [K] for similar presentation).
If we change our Weyl chamber, we can repeat this construction, defining first anti semi standard tableaux as Young tableaux with entries strictly decreasing in each column and decreasing in each row. Then we define an ordering on the set of variables δ s i 1 ,...,is , i 1 > i 2 > · · · > i s by putting: δ (1) > δ (2) > . . . and δ Let T be an anti semi standard tableau. We can associate to T a monomial: and exchange the variables corresponding to columns with equal height, then we get another Young tableau T ′ such that δ T = δ T ′ . For instance: Unfortunately, if n > 2, T ′ is generally not semi standard (T = 3 1 2 , T ′ = 2 1 3 ) thus our change of ordering on the variables δ defines a new Groebner basis on the shape algebra if n > 2. Now, the symmetry τ corresponds to the following operation on tableaux since: We can define τ directly on Young tableaux by replacing each entry a i j of T by n + 1 − a i j . The anti semi standard tableaux are exactly the image by τ of the semi standard ones.
6. The reduced shape algebra : Algebraic presentation Let V be a complex vector space with dimension n. From now one, we shall study a quotient of the shape algebra S • (V ).

Definition 2.
Let R + be the ideal in the shape algebra generated by v λ − 1: We call reduced shape algebra and write This reduced shape algebra is no more a natural sl(n) module but the ideal R is invariant under the action of the solvable group HN + consisting of upper triangula matrices in SL(n). Thus the quotient is a HN + module too. The action of the Cartan group H is still diagonal, let study the N + (or n + ) action on S • red (V ) + .
ii) Let u be a non zero vector in S • red (V ) + , the n + module W generated by u is finite dimensional since u is a finite sum of image through π + of weights vectors. The n + action is locally nilpotent on S • (V ), thus it is also locally nilpotent on S • red (V ) + , as a consequence W contains a non trivial vector annhilated by n + . This vector is a multiple of 1. Thus iii) Let π + λ be the restriction of π + to S λ (V ). It is a morphism of n + modules. If its kernel is not vanishing, thanks to Lie theorem, the n + module Ker(π + λ ) contains a non zero vector annihilated by n + , this vector is a multiple of v λ , but π + (v λ ) = 1 = 0. Thus π + λ is an isomorphism of n + modules.
iv) The relation λ > µ is equivalent to say there is ν dominant integral weight such that λ = µ + ν. In S • (V ), the multiplication by v ν send S µ (V ) into S λ (V ). In the quotient, this operation becomes the identity mapping: π + (uv ν ) = π + (u) for any u in S µ (V ).
7. The reduced shape algebra, Geometrical presentation As above, we can write everything in term of the functions δ .
Suppose now f is a polynomial function, invariant with respect to the right multiplication by N + . Then f is characterized by its restriction to the dense open subset of SL(n) of the matrices g such that δ (p) 1,...,p (g) = 0 for all p. On this set, by the use of the Gauss method, we can write: With, for all k ≥ j: 1,2,...,k−1 (g) .
By N + right invariance, we get By definition, the functions Φ c 1 ,...,c n−1 and F c 1 ,...,c n−1 are polynomial, F c 1 ,...,c n−1 is right invariant by N + and The restriction of the function f to N − characterizes the function Φ 0,...,0 thus the value of F 0,...,0 and F 0,...,0 and f are in the same class modulo R(δ) + . Conversely, any polynomial function Φ on N + defines a function F in C[SL(n, C)] N + . The restriction mapping is an isomorphism of algebra between S • red (V ) + and C[N − ].

Remark 3.
In this presentation of S • red (V ) + , the N + action on the elements of the reduced shape algebra is very natural since it is just: , we have also: Where P red (δ) + is the ideal generated by the Plücker relations but where we replace the function δ 1,...,j by 1.
And any f is modulo R(δ) − characterized by its restriction to: Theorem 3.
The reduced shape algebras are isomorphic to the algebra of polynomial functions on N − (N + ), then: The last assertions of the theorem comes from the observation that the exponential mapping from the Lie algebra n − (n + ) onto the Lie group N − (N + ) is a polynomial bijection with inverse polynomial too.
8. The reduced shape algebra: Combinatorial presentation 8.1. Super and quasi standard Young tableaux.
In order to describe the restricted shape algebra and the restricted Plücker relations, we have to perform the quotient of the preceding construction by the ideal generated by {δ A column whose height is c in a tableau is trivial if its entries are 1, 2, . . . , c, a Young tableau T is trivial if each column of T is trivial. Now let T be a Young tableau (semi standard or not), we define the extraction of trivial columns in T in the following manner: Denote a ij the entries of T (a ij is in the row i and the column j, for any j, a ij < a (i+1)j and the heights c 1 , . . . , c t of T are decreasing). We say that the tableau T is reducible if • there is a column j whose the s top entries are 1, 2, . . . , s (a i,j = i for 1 ≤ i ≤ s), • on the right of the column j, there is a column j ′ with height s in T (there is j ′ ≥ j such that c j ′ = s), • for any k > j, if c k−1 > s and c k ≥ s, a s+1,k−1 > a s,k . Let T be a reducible Young tableau, let j the smallest index and s the largest index for which the above conditions hold. Let us suppress the trivial top part of the column j and shift to the left the right parts of the s first rows (i.e. we shift to the left every a ik with 1 ≤ i ≤ s and j < k), then we get a Young tableau R 1 : the entries of If the number of column of T was t, then R 1 has t − 1 column, more precisely if the heights of the columns of T were: (c 1 , . . . , c t ) and the columns of heights s had the number j ′ , . . . , j ′′ , then the heights of the Simultaneously, we define L 1 as the Young tableau with only one trivial column with entries 1, . . . , s.
Now if R 1 is reducible, we repeat the above operation, extracting a second trivial column from R 1 , getting two Young tableaux a trivial one with two columns L 2 and a Young tableau R 2 with t − 2 columns.
Repeating this construction, after m steps, we get a trivial Young tableau L m with m column and a Young tableau R m with t−m columns.
This construction stops when the Young tableau R m is not reducible we say R m is irreducible and call R m the residue of T .

Definition 3. (Super, left and right Young tableaux)
A super Young tableau is a pair S = (L, R) of two Young tableaux, the left one L is a trivial Young tableau, the right one, R is an irreducible Young tableau. L or R can be the empty tableau without any column.
Our construction defines a mapping f (the extraction mapping) from the set Y of Young tableaux into the set SY of super young tableaux f (T ) = S = (L, R).
If λ is the sequence of the heights of the column of T : λ = (c 1 ≤ · · · ≤ c t ) and µ = (c ′ 1 ≤ · · · ≤ c ′ ℓ ) and ν = (c ′′ 1 ≤ · · · ≤ c ′′ r ) the corresponding sequence for L and R (one of these sequences can be empty), then µ and ν are two disjoint subsequences of λ and λ is the 'union' of µ and ν: we refind the sequence λ by putting together the elements of µ and ν and ordering them in a decreasing sequence.
Starting with an irreducible Young tableau R, we can insert to it any family of trivial columns, say {D 1 , . . . , D ℓ }, getting a new tableau T . We insert these column in the following way: if the height of D i is d i , we insert D 1 , . . . , D i such that any column of T , after D i has height strictly less then d i , the columns of T before D i are the columns of R with length at least d i , with their ordering and the column D j (j < i). Then T is a Young tableau. Of course, if ℓ > 0, T is reducible.
Let us try to extract a trivial column from T . Among the new column, the first one is D 1 with height d 1 . In T this column is the column p. Suppose the first trivial column extracted from T is the s top elements of the column j, with j < p. Since R is irreducible, there is a k > j such that c R k−1 > s, c R k ≥ s and a R s+1,k−1 ≤ a R s,k (we denote c R k the height of the column k and a R i,j the i, j-entry in R). We choose the smallest such k. Since we can now extract the trivial column from T , there is, in T , at least one new column, say D between the two columns k − 1, k in R, which are now columns k 1 , k 2 in T . We choose for D the last one: D is the column We cannot extract the trivial column consisting of the s top elements of the column j, with j < p. Of course, we can extract all the column p of T . Thus, in the computing of f (T ), the first step is just to eliminate the column D 1 from T , repeating this construction, we get f (T ) = (L, R) where L is the trivial tableau (D 1 , . . . , D ℓ ). We proved that f is a surjective mapping by defining a mapping h from SY to Y such that f • h(L, R) = (L, R).

Definition 4. (Quasi standard tableaux)
A super Young tableau S = (L, R) is said quasi-standard if its right tableau R is semistandard.
A Young tableau T is said quasi-standard if it is irreducible and semistandard.
Let us denote by QSY (resp. QY) the set of quasi standard super Young tableaux (resp. quasi standard Young tableaux). Denote SEM the set of semistandard Young tableaux.

Lemma 1. (f is a bijection from SEM onto QSY)
The mapping f , when restricted to SEM is a one-to-one onto mapping from SEM onto QSY.

Proof
First it is clear that if T is semistandard, then each tableau in the sequence R 1 , . . . , R m defined above is still semistandard, then f is a map from SEM to QSY. Now let S = (L, R) be an element of QSY. Denote the rows of L by (L ′ 1 , . . . , L ′ u ), their lengths being ℓ ′ 1 , . . . , ℓ ′ u . Similarly, denote (L ′′ 1 , . . . , L ′′ v ) the rows of R, their lengths being ℓ ′′ 1 , . . . , ℓ ′′ v .We define the new tableau T = g(S) as the tableau with the row i contains (from left to right) ℓ ′ i entries i, then the ℓ ′′ i entries of the row i of R. In fact, T is a Young tableau since if a T i,j is an entry of T , it is either i or an entry T is semistandard: by construction each row in T is a increasing sequence of entries. g is a map from QSY to SEM.
The map g is the inverse mapping of f . Indeed if T is semistandard, if a column C of T begins by a trivial part, then all the columns before C begin with the same trivial part and suppressing the top of the first column or the top of C is the same operation, thus to construct the sequence R 1 , . . . , R m , we just have to consider the first column at each step.
Starting with T = g(S), we can extract at each step a trivial column having the height of the corresponding column of L, but no more, since R is irreducible. Thus f • g(S) = S, for any S ∈ QSY.
Conversely, starting with a semistandard T , we build first f (T ) = (L, R) and by construction the rows of L are the left part of the rows of T , thus g • f (T ) = T .

Quasi standard Young tableaux and Groebner basis.
In this section we shall repeat the construction of section 3 but for the ideal R(δ) + and the quasi standard Young tableaux.
First, we choose the following elimination order on the variables δ: defining the degree deg(δ ..is is trivial), the degree of δ T is the sum of degree of each variables and T > T ′ if and only if: and T > T ′ for the preceding ordering. Now we look for the leading terms of elements of R(δ) + , for this ordering. We saw that the leading terms of elements of P (δ) for the preceding ordering were non semistandard monomials.
Let T be a non quasi standard tableau.
Case 1: T is non semi standard.
Then T contains a non semistandard tableau with two columns T 0 : δ T = δ U δ T 0 . For T 0 , we saw there is a Plücker relation P T 0 in P (δ) whose leading term for the ordering of section 3 was T 0 .
Case 1.1: T 0 contains a trivial column C i , since T 0 is non semistandard, it is its second column. δ T 0 = δ with c ≥ s, there is j such that a j > b j , we choose the largest such j, due to our conventions of writing, if c = s then j < s and a c > c, b s > s.
Thus the relation P T 0 has the following form: If a tableau S in this relation contains a trivial column, i.e S = C 1 C 2 with C 1 trivial, we replace S by C 2 since Repeating this operation, we get an element Case 2: T is semi standard.
If T has only one column, this column is trivial T is the leading term of some P T = δ T − 1 in R(δ) + .
Since T is semi standard the construction of the super Young tableau f (T ) begins with the extraction of the top s elements 1, . . . , s of the first column of T . Let us look to the two first columns of T , C T 1 and C T 2 . By hypothesis, δ C T 1 = δ (c 1 ) 1,...,s,a s+1 ,...,ac 1 , δ C T 2 = δ (c 2 ) b 1 ,...,bs,b s+1 ,...,bc 2 and b s < a s+1 .
Let us define ∂T as the tableau with the following first columns C ∂T 1 and C ∂T 2 : Each term δ T A in the sum has a second column containing a i with i > s, thus a i ≥ a s+1 > b s and δ , δ T is the leading term of an element in R(δ) + . If c 2 > s, we repeat this construction for ∂T , forgotting its first column. We get the following element of R(δ) + : B∪{c s+1 ,...,cc 3 } δ (c 2 ) ({1,...,s,b s+1 ,...,bc 2 }\B)∪{c 1 ,...,cs} δ C ∂T Each term δ T B in the sum has a third column containing b i with i > s, B∪{c s+1 ,...,cc 3 } < δ C T 3 , δ T B < δ T . Repeating this operation we finally get an element in R(δ) + of the form: and δ T is the leading term of an element of R(δ) + .

The tableau ∂ k T considered here is (perhaps up a reordering of the columns with height s ) the tableau h(C, R 1 ) if C is the first trivial column:
δ C = δ (s) 1...s and R 1 the first step in the process of trivial columns extraction from T .
We got an element of R(δ) + : If R 1 is quasi standard, we stop the process. If it is not the case, we continue the extraction, getting new tableaux T ′ k < R 1 < T . Finally we get: T = g(L, R) with L = ∅ and δ T − δ R − T k <T a k δ T k belongs to R(δ) + . R is quasi standard R < T, T k < T . We repeat this operation for each non quasi standard T k ; getting an element P We proved that each non quasi standard Young tableau is the leading term of an explicit element P T of R(δ) + . Let us now prove that any quasi standard Young tableau is not a leading term of an element in R(δ) + .
Let λ be a highest weight for sl(n) and V λ the corresponding simple module. We saw that V λ is naturally a sub-module of S • red (V ). More precisely, V λ is the space spanned by the classes modulo R(δ) + of the monomials δ T for all Young tableau T of shape λ. A basis for V λ is given by the classes of the monomials δ T for T semi standard with shape λ in the quotient C[δ]/R(δ) + . Let us consider the sub-space W λ of V λ spanned by the quasi standard and semi standard Young tableau of shape λ. A basis of V λ is given by the classes of δ T , T semi standard with shape λ modulo R(δ) + . Either T is quasi standard and (δ T ) is a basis of W λ or T = g(L, R), we saw δ T − δ R = a k δ T k modulo R(δ) + with T k quasi standard T k < T and R is quasi standard with shape µ < λ. This proves that V λ is a subspace of µ≤λ W µ .But since g is injective, Let now T be a quasi standard Young tableau of shape λ. Suppose δ T is the leading term of an element T + k a k δ T k in R(δ) + , then using the first part of the proof, we can replace each δ T k with a non quasi standard T k , by a linear combination of δ T j with quasi standard T j modulo R(δ) + . Finally we get an element in R(δ) + of the form T + j a j δ T j with any T j quasi standard and strictly smaller than T . The shape µ j of T j is thus smaller than λ. But this is impossible since the sum µ≤λ W µ is direct.
Finally, as in section 3, for each non quasi standard Young tableau, we got an element in R(δ) + of the form: with δ T j strictly smaller than δ T and quasi standard.
Let T be a non quasi standard tableau with shape λ. We shall say that T is minimal if it does not contain any non quasi standard tableau with shape µ < λ. For instance a semi standard non quasi standard tableau with one column or with 2 columns without trivial column are minimal.
If n ≤ 3 there are no other semi standard, minimal, non quasi standard tableaux, but if n ≥ 4 there is semi standard, minimal, non quasi standard tableau with 3 columns for instance: 1 2 3 3 4

Theorem 4. The Groebner basis
The set G = {P T red , T semi standard minimal non quasi standard or T non semi standard with 2 columns, without any trivial column} is the reduced Groebner basis of R(δ) + for our ordering.

Proof:
We saw that If T contains a trivial column C, δ T is divisible by δ C and C is minimal semi standard non quasi standard.
If T is non semi standard, it contains a non semi standard tableau S with 2 columns, without any trivial column.
If T does not contain any trivial column and is semi standard then by definition it contains a minimal non quasi standard tableau S but S is by construction semi standard.

Thus
< LT (G) >=< LT (R(δ) + ) > . Now each monomial in any P T red of G which are not the leading term, are a T ′ δ T ′ with T ′ quasi standard.
But if S ⊂ T ′ , then S is also quasi standard. Indeed, S is semi standard, suppose S non quasi standard then S contains a first column C 1 = (1, 2, . . . , s, a s+1 , . . . , a C 1 ), . . , c ct ) with t ≤ s. We can extract (1, 2, . . . , s) from S. Now, we can refind T from S by adding some columns before C 1 , between columns of S or after C t . But T is semi standard. By considering each case for these new columns, we directly see that the top (1, 2, . . . , s) of columns C 1 can still be extracted from T which is impossible since T is quasi standard.
Thus any monomial of P T red is not divisible by the leading term of another P T red .
This means that G is the reduced Groebner basis of R(δ) + for our ordering.
The same result holds with the anti standard tableau, image by τ of the quasi standard tableaux.
The anti quasi standard tableaux can be defined exactly as the quasi standard tableaux by extracting "trivial" top of columns like: They are still the image by τ of the quasi standard tableaux.

Remark 5.
In fact, if n ≤ 3, the quasi satndard Grobner basis is invariant under the action of θ. Similarly, with the symmetry τ , if we identify τ (T ) with ±T ′ with T ′ the Young tableau such that δτ (T ) = δT ′ , then T quasi standard implies T ′ quasi standard. In the study of sl(4) below, we shall see this no more true for n > 3.
Let us now picture the adjoint representation of sl (3)   We resume our construction by the two following diagrams: 9. The sl(2) case 9.1. Representations of sl(2). The sl(2)-simple modules are charachterized by a highest weight a. More precesely, the basis of sl(2) is: If a is a positive integer, the simple module π a acting on the space V a is a + 1-dimensional, with a basis v n (0 ≤ n ≤ a) and the matrices of the action are: There is only one fundamental representation, associated to the weight ω 1 . We realize it in the space generated by the functions δ (1) 1 (g) = g 11 , δ (1) 2 (g) = g 21 . The other representations are realized on the space of homogeneous polynomial functions of degree a in these variables.
9.2. Shape and reduced shape algebra.
There are no Plücker relation between g 11 and g 22 , thus the shape algebra is isomorphic to the algebra The reduced shape algebra is the quotient by the ideal generated by g 11 − 1. Let us put: Then: , The X α acts on a polynomial function as the operator: We realize the sl(2)-diamond cone as the half line of the entire nodes 0, 1, . . . , a, a + 1, . . . , at each node n, we put the quasi standard Young tableau 2 . . . 2 or the monomial X n . We have an explicit basis for the representation of N + on the diamond cone defined by the action of X α , pictured by the graph: For any a ≥ 0, we define the diamond D a as the graph generated by X a , the vector space V a as the vector space with basis the nodes of D a . We saw that the anti semi standard (resp. the anti quasi standard) basis can be identified with the semi standard (resp. the quasi standard) basis. More precisely, a being fixed, the action of τ on V a , denoted by τ (a) is defined as: We can see τ (a) as the succession of the operations: We put: and H α = [X α , Y α ] or: H α (X n ) = [(n + 1)(a − n) − n(a − n + 1)]X n = (a − 2n)X n .
We complete the diamond D a by adding the edges corresponding to the Y α -action. The sl(3)-simple modules are characterized by their highest weight. More precisely, the basis of sl(3) is: The simple modules have non multiplicity free weights. We can describe then by using the reduced shape algebra. The fundamental modules are three dimensional, they are realized on the space V ω 1 = C 3 and V ω 2 = ∧ 2 C 3 . For each pair of natural integers, there is an unique irreducible representation π(a, b) with highest weight a̟ 1 + b̟ 2 .
The reduced shape algebra is obtained by imposing δ (1) 1 = 1 and δ (2) 12 = 1. An explicit description of a basis for this module V (a,b) and the X η , Y η , H η actions on this basis can be found in [W] for instance. More precisely, Wildberger defines a diamond cone D in R 3 and a infinite dimensional vector space V with basis: He defines the action of X η on these vectors e m,n,ℓ and the irreducible module V (a,b) with highest weight a̟ 1 + b̟ 2 is the module generated by the X η action on the highest weight vector e a+b,a+b,a−b . A basis for this module is an explicit subset B (a,b) of B. There is a symmetry τ (a,b) on V (a,b) , τ (a,b) (B (a,b) ) = B (a,b) and the Y η , H η actions are defined as: see [W] for explicit formulas.
Then the leading term for this basis is δ (1) 2 δ (2) 13 = XY , thus we get the basis: 2 ) y = U u E e Y y , u, e, y ∈ N, y > 0 . Now the action of X α , X β and X α+β on these polynomials are the following: Then the X η acting by derivations on the polynomial functions φ, we refind the diamond cone, the diamond D (a,b) , the vector space V (a,b) , the symmetry τ (a,b) and the complete diamond graphs on D (a,b) described in [W] with the identification: 10.3. X η action, Symmetry and Y η action.
As above, we have simple roots α, β and γ, with: Moreover we have positive roots α + β, β + γ and α + β + γ, with: We put Y η = t X η and The fundamental representations are 4 and 6 dimensional, they are associted to the fundamental highest weight ω 1 for the canonical representation on V = C 4 , ω 2 for the representation on ∧ 2 V and ω 3 for the representation on ∧ 3 V . These fundamental representations are easy to describe, the reduction of the tensor product of any two of them is completely described in [FH]. Especially, we get the Plücker relations via this decomposition.
Now the basis of our space, i.e. the nodes of the sl(4)-diamond are monomials The action of our generators X α , X β and X γ on these polynomials are: Then we get: Thus the X η for η simple are acting on our basis of the reduced shape algebra by giving linear combination with integral coefficients, indeed, we find first such a linear combination on Z (even Z + ) coefficients but on monomials which are perhaps not all admissible, then we come back to admissible monomials, using the reduced Plücker relations, but these relations are with coefficients ±1, thus we finally get a combination of monomials in the basis with coefficients in Z.

Symmetry.
Now the symmetry τ on Young tableaux does not induce a simple operation τ (abc) on the basis of the simple module V (abc) . The matrix of Y τη on this new basis is exactly the matrix of X η in the old one.