# Standard Monomial basis for other types

For the algebraic group $SL_n$ (type $A_{n-1}$) and for a dominant weight $\lambda$ the standard monomials are indexed by the semi-standard young tableaux of shape $\lambda$ and they form a basis for the representation $V_{\lambda}^*$. For other types of simple algebraic groups do we have a description of standard monomials in terms of tableau ? If so, then how are the semi-standard young tableaux described ? A small example illustrating the basis elements say for $B_4$ or $D_5$ in terms of tableau will be highly appreciated.

• A small comment: your tag 'lie-groups' is much less appropriate here than 'algebraic-groups'. – Jim Humphreys Apr 8 '17 at 19:48

Standard monomial theory has been extended to all classical groups by Lakshmibai, Seshadri and others in the series of papers "Geometry of $G/P$ I-IX".
• It looks like the standard tableaux he defined for type $B,C,D$ in the appendix is not an indexing set for a basis of the corresponding representation. I tried with some small example and observed that one needs to have more generators. He also never claimed so in this paper. Is there any other reference where one can find such a description ? – Mark Shiffor Apr 10 '17 at 15:25
• Part (b) of the Theorem on p. 366 describes the restriction of an irreducible $G$-representation $V$ to a Levi subgroup $L$. If $L=T$, the maximal torus, then it says that $G$-standard Young tableau of shape $p(\lambda)$ are indexing a basis of $V$. Is that wrong? – Friedrich Knop Apr 10 '17 at 19:02