Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 September 2013

Part I

3. Answers for $GL(n,ℂ),$ the general linear group

The results for the general linear group are just as beautiful and just as fundamental as those for the symmetric group. The results are surprisingly similar and yet different in many crucial ways. We shall see that the results for $GL(n,ℂ)$ have been generalized to a very wide class of groups whereas the results for $S_n$ have only been generalized successfully to groups that look very similar to symmetric groups. The representation theory of $GL(n,ℂ)$ was put on a very firm footing from the fundamental work of Schur [Sch1901,Sch1927] in 1901 and 1927.

I. What are the irreducible $GL(n,ℂ)\text{-modules?}$

(a)	How do we index/count them?
	There is a bijection $$\text{Partitions}\hspace{0.17em}\lambda \hspace{0.17em}\text{with at most}\hspace{0.17em}n\hspace{0.17em}\text{rows}\stackrel{1-1}{⟷}\text{Irreducible polynomial representations}\hspace{0.17em}{V}^{\lambda }\hspace{0.17em}\text{.}$$ See Appendix A4 for a definition and discussion of what it means to be a polynomial representation.
(b)	What are their dimensions?
	The dimension of the irreducible representation $V^{\lambda }$ is given by $$\begin{array}{ccc}\text{dim}\left(V^{\lambda }\right)& =& \text{\# of column strict tableaux of shape}\hspace{0.17em}\lambda \hspace{0.17em}\text{filled with entries from}\hspace{0.17em}\{1,2,\dots ,n\}\\ & =& \prod _{x\in \lambda }\frac{n+c\left(x\right)}{h_x},\end{array}$$ where $c\left(x\right)$ is the content of the box $x$ and $h_x$ is the hook length at the box $x\text{.}$
(c)	What are their characters?
	Let ${\chi }^{\lambda }\left(g\right)$ be the character of the irreducible representation $V^{\lambda }$ evaluated at an element $g\in GL(n,ℂ)\text{.}$ The character ${\chi }^{\lambda }\left(g\right)$ is given by $$\begin{array}{ccc}{\chi }^{\lambda }\left(g\right)& =& \sum _T{x}^T\\ & =& \frac{{\sum }_{w\in S_n}\epsilon \left(w\right)w{x}^{\lambda +\delta }}{{\sum }_{w\in S_n}\epsilon \left(w\right)w{x}^{\delta }}=\frac{\text{det}\left(x_i^{{\lambda }_j+n-j}\right)}{\text{det}\left(x_i^{n-j}\right)},\end{array}$$ where the sum is over all column strict tableaux $T$ of shape $\lambda$ filled with entries from $\{1,2,\dots ,n\}$ and $$x^T=x^{{\mu }_1}{x}_2^{{\mu }_2}\cdots x_n^{{\mu }_n},\text{where}\hspace{0.17em}{\mu }_i=\text{\# of}\hspace{0.17em}i\text{'s in}\hspace{0.17em}T$$ and $x_1,x_2,\dots ,x_n$ are the eigenvalues of the matrix $g\text{.}$ Let us not worry about the first expression in the second line at the moment. Let us only say that it is routine to rewrite it as the second expression in that line which is one of the standard expressions for the Schur function, see [Mac1995] I §3.

C. How do we construct the irreducible modules?

There are several interesting constructions of the irreducible $V^{\lambda }\text{.}$

(C1)	via Young symmetrizers.
	Recall that the irreducible $S^{\lambda }$ of the symmetric group $S_k$ was constructed via Young symmetrizers in the form $$S^{\lambda }\cong ℂS_{n}P\left(T\right)N\left(T\right)\text{.}$$ We can construct the irreducible $GL(n,ℂ)\text{-module}$ in a similar form. If $\lambda$ is a partition of $k$ then $$V^{\lambda }\cong V^{\otimes k}P\left(T\right)N\left(T\right)\text{.}$$ This important construction is detailed in Appendix A5.
(C2)	Gelfan’d-Tsetlin bases
	This construction of the irreducible $GL(n,ℂ)$ representations $V^{\lambda }$ is analo-gous to the Young’s seminormal construction of the irreducible representations $S^{\lambda }$ of the symmetric group. Let $$V^{\lambda }=\text{span-}\left\{v_T\hspace{0.17em}\right|\hspace{0.17em}T\hspace{0.17em}\text{are column strict tableaux of shape}\hspace{0.17em}\lambda \hspace{0.17em}\text{filled with elements of}\hspace{0.17em}\{1,2,\dots ,n\}\}$$ so that the vectors $v_T$ are a basis of $V^{\lambda }\text{.}$ Define an action of symbols $E_{k-1,k},$ $2\le k\le n,$ on the basis vectors $v_T$ by $$E_{k-1,k}{v}_T=\sum _{T^{-}}{a}_{T^{-}T}\left(k\right)v_{T^{-}},$$ where the sum is over all column strict tableaux $T^{-}$ which are obtained from $T$ by changing a $k$ to a $k-1$ and the coefficients $a_{T^{-}T}\left(k\right)$ are given by $$a_{T^{-}T}\left(k\right)=-\frac{\prod _{i=1}^k(T_{ik}-T_{j,k-1}+j-k)}{\prod _{\underset{i\ne j}{i=1}}^{k-1}(T_{i,k-1}-T_{j,k-1}+j-k)},$$ where $j$ is the row number of the entry where $T^{-}$ and $T$ differ and $T_{ik}$ is the position of the rightmost entry $\le k$ in row $i$ of $T\text{.}$ Similarly, define an action of symbols $E_{k,k-1},$ $2\le k\le n,$ on the basis vectors $v_T$ by $$E_{k,k-1}{v}_T=\sum _{T^{+}}{b}_{T^{+}T}\left(k\right)v_{T^{+}},$$ where the sum is over all column strict tableaux $T^{+}$ which are obtained from $T$ by changing a $k-1$ to a $k$ and the coefficients $b_{T^{+}T}\left(k\right)$ are given by $$b_{T^{+}T}\left(k\right)=\frac{\prod _{i=1}^{k-2}(T_{ik-2}-T_{j,k-1}+j-k)}{\prod _{\underset{i\ne j}{i=1}}^{k-1}(T_{i,k-1}-T_{j,k-1}+j-k)},$$ where $j$ is the row number of the entry where $T^{+}$ and $T$ differ and $T_{ik}$ is the position of the rightmost entry $\le k$ in row $i\text{.}$ Since $$\begin{array}{c}{g}_i\left(x\right)=\left(\begin{array}{ccccccc}1& 0& & \cdots & & & 0\\ 0& \ddots \\ & & 1& & & & ⋮\\ ⋮& & & z\\ & & & & 1\\ & & & & & \ddots & 0\\ 0& & & \cdots & & 0& 1\end{array}\right),z\in {ℂ}^{*},\\ g_{i-1,i}\left(z\right)=\left(\begin{array}{cccccc}1& 0& \cdots & & & 0\\ 0& \ddots \\ & & 1& z& & ⋮\\ ⋮& & 0& 1\\ & & & & \ddots & 0\\ 0& & \cdots & & 0& 1\end{array}\right),z\in ℂ,\\ g_{i,i-1}\left(z\right)=\left(\begin{array}{cccccc}1& 0& \cdots & & & 0\\ 0& \ddots \\ & & 1& & & ⋮\\ ⋮& & z& 1\\ & & & & \ddots & 0\\ 0& & \cdots & & 0& 1\end{array}\right),z\in ℂ,\end{array}$$ generate $GL(n,ℂ),$ the action of these matrices on the basis vectors $v_T$ will determine the action of all of $GL(n,ℂ)$ on the space $V^{\lambda }\text{.}$ The action of these generators is given by: $$\begin{array}{ccc}{g}_i\left(z\right)v_T& =& z^{\left(\text{\# of}\hspace{0.17em}i\text{'s in}\hspace{0.17em}T\right)}{v}_T,\\ g_{i-1,i}\left(z\right)v_T& =& e^{z{E}_{i-1,i}}{v}_T=(1+z{E}_{i-1,i}+\frac{1}{2!}{z}^2{E}_{i-1,i}^2+\dots )v_T,\\ g_{i,i-1}\left(z\right)v_T& =& e^{z{E}_{i,i-1}}{v}_T=(1+z{E}_{i,i-1}+\frac{1}{2!}{z}^2{E}_{i,i-1}^2+\dots )v_T\text{.}\end{array}$$
	$b(S,T)$ is the box where $S$ and $T$ differ, $r\left(b(S,T)\right)$ is the row number of the box $b(S,T),$ $p\left(b(S,T)\right)$ is the position of the box $b(S,T)$ in its row, $p(\le k,i)$ is the position of the rightmost entry.
(C3)	The Borel-Weil-Bott construction
	Let $\lambda$ be a partition. Then $\lambda$ defines a character (one-dimensional representation) of the group $T_n$ of diagonal matrices in $G=GL(n,ℂ)\text{.}$ This character can be extended to the group $B=B_n$ of upper triangular matrices in $G=GL(n,ℂ)$ by letting it act trivially on $U_n$ the group of upper unitriangular matrices in $G=GL(n,ℂ)\text{.}$ Then the fiber product $${ℒ}_{\lambda }=G{\times }_B\lambda$$ is a line bundle on $G/B\text{.}$ Finally, $$V^{\lambda }\cong H^0(G/B,{ℒ}_{\lambda }),$$ where $H^0(G/B,{ℒ}_{\lambda })$ is the space of global sections of the line bundle ${ℒ}_{\lambda }\text{.}$ More details on the construction of the character $\lambda$ and the line bundle ${ℒ}_{\lambda }$ are given in Appendix A6.

S. Special/Interesting representations

(S1)

Let $$GL\left(k\right)\times GL\left(\ell \right)=\left(\begin{array}{cc}\left(\begin{array}{c}\\ GL(k,ℂ)\\ \end{array}\right)& 0\\ 0& \left(\begin{array}{c}\\ GL(k,ℂ)\\ \end{array}\right)\end{array}\right)\subseteq GL\left(n\right),\text{where}\hspace{0.17em}k+\ell =n\text{.}$$ Then $$V^{\lambda }{\downarrow }_{GL\left(k\right)\times GL\left(\ell \right)}^{GL\left(n\right)}=\sum _{\mu ,\nu }{c}_{\mu \nu }^{\lambda }(V^{\mu }\otimes V^{\nu }),$$ where $c_{\mu \nu }^{\lambda }$ is the number of column strict fillings of $\lambda /\mu$ with content $\nu$ such that the word of the filling is a lattice permutation. The positive integers $c_{\mu \nu }^{\lambda }$ are the Littlewood-Richardson coefficients that appeared earlier in the decomposition of $S^{\lambda }{\downarrow }_{S_k\times S_{\ell }}^{S_n}$ in terms of $S^{\mu }\otimes S^{\nu }\text{.}$ We may write this expansion in the form $$V^{\lambda }{\downarrow }_{GL\left(k\right)\times GL\left(\ell \right)}^{GL\left(n\right)}=\sum _{F\hspace{0.17em}\text{fillings}}{V}^{\mu \left(F\right)}\otimes V^{\nu \left(F\right)}\text{.}$$ We could do this precisely if we wanted. We won’t do it now, but the point is that it may be nice to write this expansion as a sum over combinatorial objects. This will be the form in which this will be generalized later.

(S2)

Let $V^{\mu }$ and $V^{\nu }$ be irreducible polynomial representations of $GL\left(n\right)\text{.}$ Then $$V^{\mu }\otimes V^{\nu }=\sum _{\lambda }{c}_{\mu \nu }^{\lambda }{V}^{\lambda },$$ where $GL\left(n\right)$ acts on $V^{\mu }\otimes V^{\nu }$ by $g(m\otimes n)=gm\otimes gn,$ for $g\in GL(n,ℂ),$ $m\in V^{\mu }$ and $n\in V^{\nu }\text{.}$ Amazingly, the coefficients $c_{\mu \nu }^{\lambda }$ are the Littlewood-Richardson coefficients again. These are the same coefficients that appeared in the (S1) case above and in the (S1) case for the symmetric group.

Remarks

(1)	There is a strong similarity between the results for the symmetric group and the results for $GL(n,ℂ)\text{.}$ One might wonder whether there is any connection between these two pictures. There are TWO DISTINCT ways of making concrete connections between the representation theories of $GL(n,ℂ)$ and the symmetric group. In fact these two are so different that DIFFERENT SYMMETRIC GROUPS are involved. (a) If $\lambda$ is a partition of $n$ then the “zero weight space”, or $(1,1,\dots ,1)$ weight space, of the irreducible $GL(n,ℂ)\text{-module}$ $V^{\lambda }$ is isomorphic to the irreducible module $S^{\lambda }$ for the group $S_n,$ where the action of $S_n$ is determined by the fact that $S_n$ is the subgroup of permutation matrices in $GL(n,ℂ)\text{.}$ This relationship is reflected in the combinatorics: the standard tableaux of shape $\lambda$ are exactly the column strict tableaux of shape $\lambda$ which are of weight $\nu =(1,1,\dots ,1)\text{.}$ (b) Schur-Weyl duality, see §A5 in the appendix, says that the action of the symmetric group $S_k$ on $V^{\otimes k}$ by permutation of the tensor factors generates the full centralizer of the $GL(n,ℂ)\text{-action}$ on $V^{\otimes k}$ where $V$ is the standard $n\text{-dimensional}$ representation of $GL(n,ℂ)\text{.}$ By double centralizer theory, this duality induces a correspondence between the irreducible representations of $GL(n,ℂ)$ which appear in $V^{\otimes k}$ and the irreducible representations of $S_k$ which appear in $V^{\otimes k}\text{.}$ These representations are indexed by partitions $\lambda$ of $k\text{.}$
(2)	It is important to note that the word character has two different and commonly used meanings and the use of the word character in (C3) is different than in Section 1. In (C3) above the word character means one dimensional representation. This terminology is used particularly (but not exclusively) in reference to representations of abelian groups (like the group $T_n$ in (C3)). In general one has to infer from the context which meaning is intended.
(3)	The indexing and the formula for the characters of the irreducible representations is due to Schur [Sch1901].
(4)	The formula for the dimensions of the irreducibles as the number of column strict tableaux follows from the work of Kostka [Kos1882] and Schur [Sch1901]. The “hook-content” formula appears in [Mac1995] I §3 Ex. 4, where the book of Littlewood [Mac1995] is quoted.
(5)	The construction of the irreducibles by Young symmetrizers appeared in 1939 in the influential book [Wey1946] of H. Weyl. It was generalized to the symplectic and orthogonal groups by H. Weyl in the same book. Further important information about this construction in the symplectic and orthogonal cases is found in [Ber1986-2] and [KWe1993]. It is not known how to generalize this construction to arbitrary complex semisimple Lie groups.
(6)	The Gelfan’d-Tsetlin basis construction originates from 1950 [GTs1950]. A similar construction was given for the orthogonal group at the same time [GTs1950-2] and was generalized to the symplectic group by Zhelobenko, see [Zhe1987,Zeh1973]. This construction does not generalize well to other complex semisimple groups since it depends crucially on a tower $G\supseteq G_1\supseteq \cdots \supseteq G_k\supseteq \left\{1\right\}$ of “nice” Lie groups such that all the combinatorics is controllable.
(7)	The Borel-Weil-Bott construction is not a combinatorial construction of the irreducible module $V^{\lambda }\text{.}$ It is very important because it is a construction that generalizes well to all other compact connected real Lie groups.
(8)	The facts about the special representations which we have given above are found in Littlewood’s book [Lit1940].

Notes and references

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..

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