Last update: 17 September 2013
Although the foundations for generalizing the $GL(n,\u2102)$ results to all complex semisimple Lie groups and Lie algebras were laid in the fundamental work of Weyl [Wey1925-26] in 1925, it is only recently that a complete generalization of the tableaux results for $GL(n,\u2102)$ has been obtained by Littelmann [Lit1995]. The results which we state below are generalizations of those given for $GL(n,\u2102)$ in the last section; partitions get replaced by points in a lattice called ${P}^{+},$ and column strict tableaux get replaced by paths. See the Appendix A7 for some basics on complex semisimple Lie algebras.
(a) | How do we index/count them? |
There is a bijection $$\lambda \in {P}^{+}\phantom{\rule{2em}{0ex}}\stackrel{1-1}{\u27f7}\hspace{0.17em}\text{irreducible representations}\hspace{0.17em}{V}^{\lambda},$$ where ${P}^{+}$ is the cone of dominant integral weights for $\U0001d524\text{.}$ The set ${P}^{+}$ is described in Appendix A8. | |
(Ib) | What are their dimensions? |
The dimension of the irreducible representation ${V}^{\lambda}$ is given by $$\begin{array}{ccc}\text{dim}\left({V}^{\lambda}\right)& =& \text{\# of paths in}\hspace{0.17em}\mathcal{P}{\pi}_{\lambda}\\ & =& \prod _{\alpha >0}\frac{\u27e8\lambda +\rho ,\alpha \u27e9}{\u27e8\rho ,\alpha \u27e9},\end{array}$$ where $$\begin{array}{c}\rho =\frac{1}{2}\sum _{\alpha >0}\alpha \text{, is the half sum of the positive roots,}\\ {\pi}_{\lambda}\hspace{0.17em}\text{is the straight line path from}\hspace{0.17em}0\hspace{0.17em}\text{to}\hspace{0.17em}\lambda \text{, and}\\ \mathcal{P}{\pi}_{\lambda}=\left\{{f}_{{i}_{1}}\cdots {f}_{{i}_{k}}{\pi}_{\lambda}\hspace{0.17em}\right|\hspace{0.17em}1\le {i}_{1},\dots ,{i}_{k}\le n\}\text{, where}\\ {f}_{1},\dots ,{f}_{n}\hspace{0.17em}\text{are the path operators introduced in [Lit1995].}\end{array}$$ We shall not define the operators ${f}_{i}$ here (or in the appendix, see [Lit1995]), let us just say that they act on paths and they are partial permutations in the sense that if ${f}_{i}$ acts on a path $\pi $ then the result is either $0$ or another path. See Appendix A8 for a few more details. | |
(c) | What are their characters? |
The character of the irreducible module ${V}^{\lambda}$ is given by $$\begin{array}{ccc}\text{char}\left({V}^{\lambda}\right)& =& \sum _{\eta \in \mathcal{P}{\pi}_{\lambda}}{e}^{\eta \left(1\right)}\\ & =& \frac{{\sum}_{w\in W}\epsilon \left(w\right){e}^{w(\lambda +\rho )}}{{\sum}_{w\in W}\epsilon \left(w\right){e}^{w\rho}},\end{array}$$ where $\eta \left(1\right)$ is the endpoint of the path $\eta \text{.}$ These expressions live in the group algebra of the weight lattice $P,$ $\u2102\left[P\right]=\text{span}\left\{{e}^{\mu}\hspace{0.17em}\right|\hspace{0.17em}\mu \in P\},$ where ${e}^{\mu}$ is a formal variable indexed by $\mu $ and the multiplication is given by ${e}^{\mu}{e}^{\nu}={e}^{\mu +\nu},$ for $\mu ,\nu \in P\text{.}$ See Appendix A7 for more details. |
(S1) | Let $\U0001d529\subseteq \U0001d524$ be a Levi subalgebra of $\U0001d524$ (this is a Lie algebra corresponding to a subgraph of the Dynkin diagram which corresponds to $\U0001d524\text{).}$ The subalgebra $\U0001d529$ corresponds to a subset $J$ of the set $\{{\alpha}_{1},\dots ,{\alpha}_{n}\}$ of simple roots. The restriction rule from $\U0001d524$ to $\U0001d529$ is $${V}^{\lambda}{\downarrow}_{\U0001d529}^{\U0001d524}=\sum _{\eta}{V}^{\eta \left(1\right)},$$ where $$\begin{array}{c}\text{the sum is over all paths}\hspace{0.17em}\eta \in \mathcal{P}{\pi}_{\lambda}\hspace{0.17em}\text{such that}\hspace{0.17em}\eta \in {\stackrel{\u203e}{C}}_{\U0001d529}\text{,}\\ \eta \in {\stackrel{\u203e}{C}}_{\U0001d529}\hspace{0.17em}\text{means that}\hspace{0.17em}\u27e8\eta \left(t\right),{\alpha}_{i}\u27e9\ge 0\text{, for all}\hspace{0.17em}t\in [0,1]\hspace{0.17em}\text{and all}\hspace{0.17em}{\alpha}_{i}\in J\text{.}\end{array}$$ |
(S2) | The tensor product of two irreducible modules is given by $${V}^{\mu}\otimes {V}^{\nu}=\sum _{\eta}{V}^{\mu +\eta \left(1\right)},$$ where the sum is over all paths $\eta \in \mathcal{P}{\pi}_{\nu}$ such that ${\pi}_{\mu}*\eta \in \stackrel{\u203e}{C}\text{,}$ $$\begin{array}{c}{\pi}_{\mu}\hspace{0.17em}\text{and}\hspace{0.17em}{\pi}_{\nu}\hspace{0.17em}\text{are straight line paths from}\hspace{0.17em}0\hspace{0.17em}\text{to}\hspace{0.17em}\mu \hspace{0.17em}\text{and}\hspace{0.17em}0\hspace{0.17em}\text{to}\hspace{0.17em}\nu \text{, respectively,}\\ \mathcal{P}{\pi}_{\nu}\hspace{0.17em}\text{is as in (Ib),}\\ {\pi}_{\mu}*\eta \hspace{0.17em}\text{is the path obtained by attaching}\hspace{0.17em}\eta \hspace{0.17em}\text{to the end of}\hspace{0.17em}{\pi}_{\mu}\text{, and}\\ ({\pi}_{\mu}*\eta )\in \stackrel{\u203e}{C}\hspace{0.17em}\text{means that}\hspace{0.17em}\u27e8({\pi}_{\mu}*\eta )\left(t\right),{\alpha}_{i}\u27e9\ge 0\text{, for all}\hspace{0.17em}t\in [0,1]\hspace{0.17em}\text{and all simple roots}\hspace{0.17em}{\alpha}_{i}\text{.}\end{array}$$ |
Notes and references
(1) | The indexing of irreducible representations given in (Ia) is due to Cartan and Killing, the founders of the theory, from around the turn of the century. Introductory treatments of this result can be found in [FHa1991] and [Hum1978]. |
(2) | The first equality in (Ib) is due to Littelmann [Lit1994], but his later article [Lit1995] has some improvements and can be read independently, so we recommend the later article. This formula for the dimension of the irreducible representation, the number of paths in a certain set, is exactly analogous to the formula in the $GL(n,\u2102)$ case, the number of tableaux which satisfy a certain condition. The second equality is the Weyl dimension formula which was originally proved in [Wey1925-26]. It can be proved easily from the Weyl character formula given in (Ic), see [Hum1978] and [Ste1994] Lemma 2.5. This product formula is an analogue of the “hook-content” formula given in the $GL(n,\u2102)$ case. |
(3) | A priori, it might be possible that the set $\mathcal{P}{\pi}_{\lambda}$ is an infinite set, at least the way that we have defined it. In fact, this set is always finite and there is a description of the paths that are contained in it. The paths in this set are called Lakshmibai-Seshadri paths, see [Lit1995]. The explicit description of these paths is a generalization of the types of indexings that were used in the “standard monomial theory” of Lakshmibai and Seshadri [LSe1989]. |
(4) | The first equality in (Ic) is due to Littelmann [Lit1995]. This formula, a weighted sum over paths, is an analogue of the formula for the irreducible character of $GL(n,\u2102)$ as a weighted sum of column strict tableaux. The second equality in (Ic) is the celebrated Weyl character formula which was originally proved in [Wey1925-26]. A modern treatment of this formula can be found in [BtD1985], [Hum1978], and [Var1984]. |
(5) | The general restriction formula (S1) is due to Littelmann [Lit1995]. This is an analogue of the rule given in (S1) of the $GL(n,\u2102)$ results. In this case the formula is as a sum over paths which satisfy certain conditions whereas in the $GL(n,\u2102)$ case the formula is a sum over column strict fillings which satisfy a certain condition. |
(6) | The general tensor product formula in (S2) is due to Littelmann [Lit1995]. This formula is an analogue of the formula given in (S2) of the $GL(n,\u2102)$ results. |
(7) | The results of Littelmann given above are some of the most exciting results of combinatorial representation theory in recent years. They were very much inspired by some very explicit conjectures of Lakshmibai, see [LSe1989], which arose out of the “standard monomial theory” developed by Lakshmibai and Seshadri. Although Littelmann’s theory is actually much more general than we have stated above, the special set of paths $\mathcal{P}{\pi}_{\lambda}$ used in (Ib-c) is a modified description of the same set which appeared in Lakshmibai’s conjecture. Another important influence on Littelmann in his work was Kashiwara’s work on crystal bases [Kas1990]. |
This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.
Key words and phrases. Algebraic combinatorics, representations.
Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..