1.5 Super algèbres de Lie
1.5.4 Relations de Garnir
Le module de forme S est le quotient de l'algèbre symmétrique S(⊕S(ωr)) par l'idéal
engendré par les relations de Garnir. Décrivons ces relations.
Soient C = (v1, . . . , vP) et D = (w1, . . . , wQ) deux suites nies de vecteurs dans
V0⊕ V1. Nous supposons que P ≥ Q et nous posons :
C ∨ D = (v1, . . . , vP, w1, . . . , wQ) = (u1, . . . , uP +Q).
Soient uC, uD et uC∨D les vecteurs de la forme
uC∨D = uC.uD = (u1∧ . . . ∧ uP).(uP +1∧ . . . ∧ uP +Q).
Pour tout σ dans le groupe des permutations SP +Q, nous dé nissons :
u(C∨D) e σ =eε σ C∨D(uσ−1(1)∧ . . . ∧ uσ−1(P )).(uσ−1(P +1)∧ . . . ∧ uσ−1(P +Q)), avec eε σ C∨D = Y 1≤a<b≤P +Q σ(a)>σ(b) (−1)|ua||ub|où |u
a|est le degré de ua eteσ est la version graduée de σ.
Dans les relations de Garnir, nous utilisons des permutations particulières σ. Soit p ≤ P, q ≤ Q, on note X = v1 ∧ . . . ∧ vp et Y = w1 ∧ . . . ∧ wq. Une sous suite à r
éléments X0 ⊂ X est une suite (v
i1, vi2, . . . , vir) telle que i1 < i2 < . . . < ir. On note
sr(X)l'ensemble de telles suites.
Si r ≤ inf(p, q) et X0 = (v
i1, vi2, . . . , vir) = (ui1, ui2, . . . , uir) dans sr(X), Y
0 =
(wj1, wj2, . . . , wjr) = (uP +j1, uP +j2, . . . , uP +jr) dans sr(Y ), on dé nit une permutation
X0 ↔ Y0 dans S
P +q par :
X0 ↔ Y0 = (i1, P + j1)(i2, P + j2) . . . (ir, P + jr)
= 1 ... i1 ... ir ... P P +1 ... P +j1 ... P +jr ... P +Q
1 ... P +j1 ... P +jr ... P P +1 ... i1 ... ir ... P +Q .
Par dé nition, la relation de Garnir sur un vecteur uC.uD associée à X et Y est :
GX,Y(uC.uD) = inf (p,q) X r=0 (−1)r X X0∈sr(X) Y0∈sr(Y ) e εXC∨D0↔Y0u(C∨D)X0↔Y 0.
King et Welsh montrent que : Théorème 1.5.1.
L'algèbre de forme de sl(m/1) est le quotient de S(⊕S(ωr)) par l'idéal engendré par
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Chapitre 2
Diamond representations for rank two
semisimples Lie algebras
B. Agrebaoui, D. Arnal and O. Khli , Published in Journal of Lie theory.
Abstract
The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor N in the Iwasawa decomposition of a semisimple Lie algebra g, using the restrictions to N of the simple nite dimensional modules of g. Such a description is given in Arnal, D., N. Bel Baraka, and N.-J. Wildberger, Diamond representations of sl(n), Annales Mathé- matiques Blaise Pascal 13 (2006), 381429 for the case g = sl(n). Here, we perform the same construction for the rank 2 semisimple Lie algebras (of type A1× A1, A2, C2 and
G2). The algebra C[N] of polynomial functions on N is a quotient, called the reduced
shape algebra, of the shape algebra for g. Bases for the shape algebra are known, for instance the so-called semi standard Young tableaux give an explicit basis (see Alverson, L.-W., R.-G. Donnelly, S.-J. Lewis, M. McClard, R. Pervine, R.-A. Proctor, and N.-J. Wildberger, Distributive lattice de ned for representations of rank two semisimple Lie algebras, SIAM J. Discrete Math. 23 (2008/09), no. 1, 527559). We select among the semi standard tableaux, the so-called quasi standard ones which de ne a kind basis for the reduced shape algebra. .
2.1 Introduction
We study the diamond cone of representations for the nilpotent factor N+ of any
rank 2 semisimple Lie algebra g. This is the indecomposable regular representation onto C[N−], described from explicit realizations of the restrictions to N+ of the simple g-
modules Vλ.
In [ABW], this description is explicitly given in the case g = sl(n), using the notion of quasi standard Young tableaux. Roughly speaking, a quasi standard Young tableau is an usual semi standard Young tableau such that, it is impossible to extract the top of the rst column, either because this top of column is not `trivial', i.e. it does not consist of numbers 1, 2, . . . , k, or because, when we extract this top by pushing to the left the k rst rows of the tableau, we do not get a semi standard tableau.
Let us come back for the case of rank 2 Lie algebra g. The modules Vλ have well
known explicit realizations (see for instance [FH]). They are characterized by their highest weight λ = aω1 + bω2, integral combination of fundamental weights. In [ADLMPPrW],
there is a construction for a basis for each Vλ, as the collection of all semi standard
tableaux with shape (a, b). The de nition and construction of semi standard tableaux for g uses the notion of grid poset and their ideals. It is possible to perform compositions of grid posets, the ideals of these compositions (of a grid posets associated to Vω1 and b
grid posets associated to Vω2) give a basis for Vλ if λ = aω
1+ bω2.
Here, we realize the Lie algebra g as a subalgebra of sl(n) (with n = 4, 3, 4, 7), and we recall the notion of shape algebra for g, it is the direct sum of all the simple modules Vλ, but we see it as the algebra C[G]N+
of all the polynomial functions on the group G (corresponding to g), which are invariant under right action by elements in N+. This
gives a very concrete interpretation of the semi standard tableaux for g as a product of determinant functions for submatrices.
The algebra C[N−]is the restriction to N− of the functions in C[G]. But it is also a
quotient of the shape algebra by the ideal generated by 12 − 1, 1 − 1. We call this quotient the reduced shape algebra for g. To give a basis for this quotient, we de ne, case by case, the quasi standard tableaux for g. They are semi standard Young tableaux, with an extra condition, which is very similar to the condition given in the sl(n) case. We prove that the quasi standard Young tableaux give a kind basis for the reduced shape algebra.
We study the diamond cone of representations for the nilpotent factor N+ of any rank 2
semisimple Lie algebra g. This is the indecomposable regular representation onto C[N−],
described from explicit realizations of the restrictions to N+ of the simple g-modules Vλ.
In [ABW], this description is explicitly given in the case g = sl(n), using the notion of quasi standard Young tableaux. Roughly speaking, a quasi standard Young tableau is an usual semi standard Young tableau such that, it is impossible to extract the top of the rst column, either because this top of column is not `trivial', i.e. it does not consist of numbers 1, 2, . . . , k, or because, when we extract this top by pushing to the left the k rst rows of the tableau, we do not get a semi standard tableau.
Let us come back for the case of rank 2 Lie algebra g. The modules Vλ have well
2.1. INTRODUCTION 43 weight λ = aω1 + bω2, integral combination of fundamental weights. In [ADLMPPrW],
there is a construction for a basis for each Vλ, as the collection of all semi standard
tableaux with shape (a, b). The de nition and construction of semi standard tableaux for g uses the notion of grid poset and their ideals. It is possible to perform compositions of grid posets, the ideals of these compositions (of a grid posets associated to Vω1 and b
grid posets associated to Vω2) give a basis for Vλ if λ = aω
1+ bω2.
Here, we realize the Lie algebra g as a subalgebra of sl(n) (with n = 4, 3, 4, 7), and we recall the notion of shape algebra for g, it is the direct sum of all the simple modules Vλ, but we see it as the algebra C[G]N+ of all the polynomial functions on the group
G (corresponding to g), which are invariant under right action by elements in N+. This gives a very concrete interpretation of the semi standard tableaux for g as a product of determinant functions for submatrices.
The algebra C[N−]is the restriction to N− of the functions in C[G]. But it is also a
quotient of the shape algebra by the ideal generated by 12 − 1, 1 − 1. We call this quotient the reduced shape algebra for g. To give a basis for this quotient, we de ne, case by case, the quasi standard tableaux for g. They are semi standard Young tableaux, with an extra condition, which is very similar to the condition given in the sl(n) case. We prove that the quasi standard Young tableaux give a kind basis for the reduced shape algebra.
We study the diamond cone of representations for the nilpotent factor N+ of any rank 2
semisimple Lie algebra g. This is the indecomposable regular representation onto C[N−],
described from explicit realizations of the restrictions to N+ of the simple g-modules Vλ.
In [ABW], this description is explicitly given in the case g = sl(n), using the notion of quasi standard Young tableaux. Roughly speaking, a quasi standard Young tableau is an usual semi standard Young tableau such that, it is impossible to extract the top of the rst column, either because this top of column is not `trivial', i.e. it does not consist of numbers 1, 2, . . . , k, or because, when we extract this top by pushing to the left the k rst rows of the tableau, we do not get a semi standard tableau.
Let us come back for the case of rank 2 Lie algebra g. The modules Vλ have well
known explicit realizations (see for instance [FH]). They are characterized by their highest weight λ = aω1 + bω2, integral combination of fundamental weights. In [ADLMPPrW],
there is a construction for a basis for each Vλ, as the collection of all semi standard
tableaux with shape (a, b). The de nition and construction of semi standard tableaux for g uses the notion of grid poset and their ideals. It is possible to perform compositions of grid posets, the ideals of these compositions (of a grid posets associated to Vω1 and b
grid posets associated to Vω2) give a basis for Vλ if λ = aω
1+ bω2.
Here, we realize the Lie algebra g as a subalgebra of sl(n) (with n = 4, 3, 4, 7), and we recall the notion of shape algebra for g, it is the direct sum of all the simple modules Vλ, but we see it as the algebra C[G]N+
of all the polynomial functions on the group G (corresponding to g), which are invariant under right action by elements in N+. This
gives a very concrete interpretation of the semi standard tableaux for g as a product of determinant functions for submatrices.
The algebra C[N−]is the restriction to N− of the functions in C[G]. But it is also a
quotient of the shape algebra by the ideal generated by 1
2 − 1, 1 − 1. We call this quotient the reduced shape algebra for g. To give a basis for this quotient, we de ne, case by case, the quasi standard tableaux for g. They are semi standard Young tableaux, with an extra condition, which is very similar to the condition given in the sl(n) case. We prove that the quasi standard Young tableaux give a kind basis for the reduced shape algebra.