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.

  • $\begingroup$ A small comment: your tag 'lie-groups' is much less appropriate here than 'algebraic-groups'. $\endgroup$ – 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".

A very concise description of standard tableaux in this setting can be found in the appendix of "Littelmann, Peter: A generalization of the Littlewood-Richardson rule. J. Algebra 130 (1990), no. 2, 328–368".

Note, that the standard monomial approach was later generalized er even superseded by Littelmann's path model and Kashiwara's crystals. These work even for symmetrizable Kac-Moody algebras.

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    $\begingroup$ For online access, note that the series of papers by Lakshmibai et al. may be harder to get, but for a short summary as of 1979 see ams.org/journals/bull/1979-01-02/S0273-0979-1979-14631-7. Littelmann's paper is also freely available: ams.org/mathscinet-getitem?mr=1051307 $\endgroup$ – Jim Humphreys Apr 8 '17 at 19:43
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    $\begingroup$ 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 ? $\endgroup$ – Mark Shiffor Apr 10 '17 at 15:25
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    $\begingroup$ 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? $\endgroup$ – Friedrich Knop Apr 10 '17 at 19:02
  • $\begingroup$ Thank you. Yes that is correct. I will check my calculations again. $\endgroup$ – Mark Shiffor Apr 10 '17 at 21:13

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