Representation TheoryIssuesFor Sage Days 7

Goals of this document� This document tries to explain a few algorithmsfrom Lie theory that would be useful to imple-ment.� I tried to include explanations of everything.There is too much to cover in my talk but I hopethat the reference will be useful.� A summary of recommended algorithms may befound at the end.

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Modules for Lie groups, algebrasHow to represent modules of a Lie group or algebra to acomputer?� Use Cartan classi�cation: Ar(�); Br(�); Cr(�);Dr(�); Er(�); F4(�); G2(�) where � is highestweight vector (HWV)� HWV itself has alternative representations.� Ring structure is important: should be able toadd or multiply:A2(1; 0) �A2(0; 1)=A2(1; 1)+A2(0; 0)gives decomposition of tensor product for SL3.� Ring structure makes dimension a homomor-phism: Dim(A2(1; 1)) should produce 8.� Branching rules are also homomorphisms.� Therefore virtual modules should be a subclass ofrings or Z-algebras.

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Weights� G a reductive complex Lie group, T max'l torus� g; t their Lie algebras� X�(T )= characters of T . Elements are weights.� If G is semisimple of rank r then X�(T )@ Zr� RX�(T )= t� is a Euclidean vector space� It is given a partial order. Positive elements arecalled dominant and form a cone, the positiveWeyl chamber C+.� Example: in A2 below �1 and �2 are simplepositive roots, "1 and "2 are fundamentalweights. The "i are dominant, �i not.

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Example: SL3=SU(3)=A2� The �nite-dimensional representations of SL3(C)are same as maximal compact subgroup SU(3).� Finite-dim'l representations of GL3(C) are thesame as U(3).� The representations of SL3 or SU(3) are almostthe same as GL3 or U(3).� G=SL3, T = ft=diag(t1; t2; t3)jt1t2t3=1g:� X�(T ) has basis with "1(t)= t1, "2(t)= t3�1.� Embed X�(T )� R3 by (�1; �2; �3)2R3�t t1�1t2�2t3�3:� For SL3, (�; �; �)� trivial character, so:� X�(T )�R3/R(1; 1; 1) or f(�1; �2; �3)jP �i=0g� "1� (1; 0; 0) or � 23 ;� 13 ;� 13�� "2� (1; 1; 0) or� 13 ; 13 ;� 23�� �1= (1;� 1; 0) and �2=(0; 1;� 1) are the simpleroots. (Picture above.)5

Highest Weight VectorsThe Weyl group W acts on weights.� The positive Weyl chamber is a fundamentaldomain for the action of W .� Restricting the character � of a representation toT it decomposes into a sum of weights �, withmultiplicities m(�)� The function m(�) is invariant under W so itssupport intersects C+.Theorem. (Weyl) Every weight in C+ is the highestweight vector (in the partial order) of a unique irre-ducible representation.Thus the representation should be parametrized by thehighest weight vector: V (�).

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A Computer NotationAssume G is semisimple of rank r� The fundamental weights "1; � ; "r are a basisof the weights (assuming G is simply-connected)that are dominant.� A special vector: � is the sum of the fundamentalweights. It is also half the sum of the positiveroots.� Use the standard labeling of "1; � ; "r in theappendix to Bourbaki, Groupes et Algebresde Lie Ch IV,V and VI.� Represent G by its Cartan classi�cation. Com-puter notation: G(n1; � ; nr) is the irreduciblemodule of G with HWV P ni"i.� Thus B3[0,0,1] is the spin module of so(7).� I'll call this Bourbaki notation.� Another notation is extensively used in the litera-ture in which highest weight vectors aredecreasing integer sequences. This notation isexplained well in Goodman and Wallach, Repre-sentations and Invariants of the Classical Groups.� We will call the notation Partition notation.We will emphasize it only for GLn but it alsoapplies to other Lie groups as a good way toexpress branching rules.7

Partition Notation� For many purposes it is su�cient to considersemisimple Lie groups.� But GLr+1 (or same: U(r + 1)) are reductive butnot semisimple.� Although they can be avoided it is better not to.� G = GLr+1, T = ft = diag(t1; � ; tr+1)g diagonaltorus.� X�(T ) @ Zr+1. In this identi�cation � 2 Zr�character t t1�1� tr�r.� Fundamental weights: "i=(1;� ; 1; 0;� ; 0), i6 r*i-th positionA dominant weight ��1>�2>� >�r:If also �r > 0 then � is a partition (of length 6 r).We'll call this �partition notation� even though if �r isnegative � is not a partition.� Partition notation is useful for branching rules.� Use partition notation for relationship betweenGLr+1 and Sk (Frobenius-Schur duality)� Partition notation is used in Symmetrica.8

Littlewood-Richardson RuleThe Littlewood-Richardson rule is implemented inSymmetrica as the Schur outer product. It manifestsitself as a homogeneous graded ring R = L Rk whereRk is the free abelian group on the partitions of k andthe multiplication Rk � Rl� Rk+l is the Littlewood-Richardson rule (LRR)� � �= X�?k+l c��� �The coe�cients c��� are nonnegative and may be de�nedcombinatorially. LRR has various meanings.� It is the tensor product rule for representations ofGLr+1.� Branching rule GL(m+n)� GL(m)�GL(n).� Induction rule Sk�Sl� Sk+l.� Structure constants for ring of symmetric polyno-mials with Schur polynomial basis.� Structure constants for cohomology of Grassman-nians with Schubert cocycle basis.

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Branching RulesIt seems important to implement branching rules forvarious inclusions of Lie groups. These are surveyed in:� Howe, Eng-Chye and Willenbring (2005).� King, J. Phys. A 8 (1975), 429�449.The most important are in the book of Goodman andWallach. Some are simple to describe, some more com-plex. If a sage framework for treating these could bearrived at for treating them by implementing an easyone or two then harder ones could be implemented atleisure. Two candidates:� The branching rule GLn � GLn�1 is particu-larly simple and important.� The branching rule GLk+l� GLk � GLl is theLittlewood-Richardson rule and could be imple-mented by exposing symmetrica.Branching rule (GLn� GLn�1). If � = (�1 >�2>� > �n) and �= (�1>� > �n�1) are highestweight vectors then the irreducible moduleVGLn(�) of GLn restricted to GLn�1 containsVGLn�1(n) if and only if � and � interleave; if soit occurs with multiplicity one.Interleaving means �1> �1>�2>� >�n.10

Frobenius-Schur DualityThere is a dictionary between the representations ofGLn and Sk. Both classes are indexed by partitions. Apartition � of k into 6n parts indexes either:� An irreducible module �Sk(�) of Sk; or� An irreducible module �GLn(�) of GLn(C).The bijection: if V =Cn then both groups act on k Vand as a bimodule it decomposeskV =M�?k �Sk(�)�GLn(�):LRR as tensor product formulaIf �; � are partitions of length 6n then�GLn(�)�GLn(�)=M� c��� ��GLn(�):LRR as branching formula�GLn+m(�)jGLn�GLm=M�;� c��� ��GLn(�)�GLm(�)LRR for symmetric groups If �?k; �?l:IndSk�SlSk+l (�Sk(�)�Sl(�))= M�?k+l c��� �Sk+l(�):

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Dimensions, weight multiplicitiesLet G be a (simply-connected) semisimple Lie group,��RX�(T ) the weight lattice. It is divided into pos-itive roots �+ and negative ones. If � is a dominantweight the Weyl dimension formula gives a fast way ofcomputing the dimension of the highest weight represen-tation V (�):dim V (�)= Y�2�+ h�; �+ �ih�; �i ; �= 12 X�2�+ �:� If the modules are implemented as a ring, this isa ring homomorphism � Z.� Mathematica:http://match.stanford.edu/bump/mma/lie.mIn[1]:= << lie.mIn[2]:= Dim[B3[0,1,0]]Out[2]= 21In[3]:= Dim[B3[0,0,1]]Out[3]= 8

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Tensor products� Branching rules are also ring homomorphisms.� We'd like the (virtual) modules over a reductiveLie group or algebra to be implemented as a Z-algebra so these homomorphisms can be handledas such. Call this Z-algebra K(G) or K(g).� If SL3=A2 we'd like A2(1,0)*A2(0,1) to returnA2(1,1)+A2(0,0)� This can be implemented by better exposing thesymmetrica function outerproduct_schur whichimplements the Littlewood-Richardson rule. Thisrequires changing Bourbaki to partition notation.A2(1,0) becomes (1,)A2(0,1) becomes (1,1)sage: symmetrica.outerproduct_schur((1,),(1,1))s[1, 1, 1] + s[2, 1]These are elements the ring of symmetric polynomials.There is a homomorphism to K(A2) in whichs[1, 1, 1] + s[2, 1]�>A2(0; 0)+A2(1; 1):� Taking dimensions should give 1+8.13

Other Lie groups� For other Lie algebras, taking the tensor prod-ucts of representations can be done e�cientlyusing a method of Brauer and Klimyk.� This requires computation of the weight multi-plicities of one of the two representations.If � is a highest weight vector, restrict the highestweight module V (�) to T and ask for the decomposi-tion: V (�)jT =X� m(�) � e�:Here e� is a synonym for � intended to make the nota-tion unambiguous. The coe�cients m(�) are essentiallygiven by the Weyl character formula but are better com-puted algorithmically using Freudenthal's multiplicityformula.� Freudenthal's formula: Humphreys, Introductionto Lie Algebras and Representation Theory Sec-tion 22.3.� Brauer-Klimyk algorithm: Humphreys, Exercise 9on page 142.� Mathematica code implementing Freudenthals'formula for A2, B2 and G2:http://match.stanford.edu/bump/mma/weights.m14

Freudenthal Multiplicity FormulaLet m(�) = m(�; �) be the multiplicity of � in highestweight module for �. Recursively:[h�+ �; �+ �i� h�+ �; �+ �i]m(�) =2 X�2�+Xi=11 m(�+ i�)h�+ i�; �i:It is useful that the support of � is known: it is(Root lattice)\ (Convex hull of W�):(W =Weyl group.) Moreover, m(�) = 1 and m is stableon W , so compute one value and you've computed up tojW j values.

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Brauer-Klimyk method for We compute V (�) V (�). The method is asymmetric.Decompose V (�)=�m(�; �) � �.� Suppose �rst � is so large that � + � 2 C+ for all� with m(�; �)� 0. ThenV (�)V (�)=M� m(�; �)V (� + �):� In the general case, discard � such that � + �+ �is on a wall of a Weyl chamber. For the survivors�nd w�2W such that w�(� + �+ �)2 C+. Thenew�(� + �+ �) is in the interior of C+ and we maywrite w�(�+ �+ �)= �0+ � where �02C+. NowV (�)V (�)=M� (� 1)l(w�)m(�; �)V (� + �0):� Since algorithm is asymmetric, choose � to be thehigher weight.

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Drawing Rank 2 Weight Diagrams� Rank 2: t�@ R2 so display the weights and multi-plicities in a diagram.

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1 A G2 weightdiagram

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An A2 weightdiagram

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1 A B2 weightdiagram17

Crystal GraphsOne may replace the Lie group G with Lie algebra g byits quantized universal enveloping algebra U(g) whichadmits a deformation Uq(g). This is a Hopf algebrawhose modules are the same as those of G but thetensor product structure is di�erent. If q� 1 it revertsto U(g) but if q� 0 then a structure emerges knownas the crystal graph.� The vertices of the crystal graph correspond tovectors in a highest weight module. Let V� bethis module and let � 2 t� be a weight. Then theweight space V�(�) is the �-eigenspace for t.� If � is a root of g and if X� � g is the eigenspacethen X� is one dimensional and a basis vector X�maps V�(�)� V�(� + �). If � is a simple rootthen we may denote the operator X� by e� andX�� by f�. Then e� (resp. f�) permute theweight spaces, mapping V�(�) � 0 if � + �(resp. ���) is not a weight.� It is not possible to choose a basis of V� that issimply permuted by the e� and f�. However bypassing to Uq(g) and (roughly) letting q � 0such a basis exists (Kashiwara). Every element ofthe crystal basis is mapped to another or to 0 bye� and f�.18

A sample crystal graphWeight diagram for A2(3; 2), and the crystal graph:1

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� It is good to draw vertices contributing to thesame weight space close together. This is possiblefor rank 2 and sometimes rank 3.� e1= e�1 and f1 shift right and left along green.� e2 = e�2 and f2 shift southeast and northwestalong red.19

Parametrizing the Crystal BasisThere are several methods of parametrizing the crystalbasis.� For Ar Gelfand-Tsetlin patterns;� For Br, Cr or Dr modi�ed �half� Gelfand-Tsetlinpatterns;� For Ar or with modi�cation Br, Cr or Dr semis-tandard Young tableaux (SSYT)� In general, string representations (Berenstein-Zelevinsky, Littelmann).� Another scheme due to Lusztig.� Historically combinatorial literature emphasizesSSYT's for everything starting with the represen-tation theory of Sk but a trend may exist toward(equivalent) Gelfand-Tsetlin patterns.� Alternating sign matrices are a special case.

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String representations� Berenstein and Zelevinsky, Canonical bases forthe quantum group of type Ar and piecewise-linear combinatorics, Duke (1996).� Littelmann, Cones, Crystals and Patterns, Trans-formation Groups (1998).� Pick a decomposition of the long element of theWeyl group into a product of simple re�ections.� This will produce an embedding of the crystalgraph into a cone in Euclidean space.� There are many decompositions if r > 3 but Lit-telmann showed that one choice leads to a conethat is describable by explicit inequalities.Deformations of the Weyl character formulaDeformations of the Weyl character formula express themodi�ed numerator as a sum over Gelfand-Tsetlin pat-terns, SSYT's, Alternating sign matrices, etc.� Older work of Tokuyama, Okada, Hamel andKing and Simpson;� More recent work of Brubaker, Bump, Chinta,Friedberg and Gunnells, a biproduct of investiga-tions motivated from automorphic forms.� These deformed Weyl character formulae may beexpressed as sums over the crystal graph and themost important data are the string data.21

Example: For A2, w0= s1s2s1.� Read this as a word telling in what order toapply the lowering operators fi.Example: Green, Red, Green.� Pick a vertex. To obtain the string, apply thegreen operator as many times x as possible; thenred as many times y as as possible; then green asmany times z as possible. At the end, you are atthe unique lowest weight vector.string data=� z yx �2 4

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� This is not the Gelfand-Tsetlin pattern. Therows increase but do not interleave.� This data uniquely determines the vertex and isa particularly useful way of representing it.22

Gelfand-Tsetlin patternsGelfand-Tsetlin patterns are a re�ection of multiplicityfree branching rules. For type A, the multiplicity-freebranching rule goes GLn� GLn�1� � . The idea isthat we can pick out an individual vector by asking thatit lie in an irreducible subspace for each branching. Inview of the branching rule described earlier, we need tospecify a partition (or decreasing integer sequence) foreach �layer� and these must interleave. The resultingpattern looks like this:�1 �2 �3 � �r�1 �2 � �r�1� ��1� The SSYT is obtained by �lling each skew parti-tion �� � etc. with an integer.8>><>>:5 3 2 14 3 14 23

9>>=>>;�1 1 1 2 42 2 33 44

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String Data and GT patternsFor the classical group, the connection between stringdata and Gelfand-Tsetlin patterns or tableaux was madeexplicit by Littelmann for a good decomposition of thelong element. Accumulate the row di�erences in the GTpattern adding the �rst several elements of one row andsubtracting the corresponding elements of the rowbelow.Gelfand-Tsetlin=

8>>>><>>>>:16 12 8 3 0+15 +9 3 1�10 �8 49 57

9>>>>=>>>>;Thus adding the red numbers and subtracting the greengives 6 which again goes in the blue spot in the �array. String Data=8>><>>:1 4 9 115 6 71 42

9>>=>>;SummaryFor any g string data parametrize of the vertices of thecrystal graph. They are also important in applications.For classical groups there are bijectionsstring data� GT patterns� tableaux24

Tensor productsIt turns out that the tensor product of two crystals hasa very simple algorithm.� Kashiwara and Nakashima, Journal of Algebra165 (1994).� A very popular algorithm explained elsewhere,e.g. Hong and Kang, Introduction to QuantumGroups and Crystal Graphs (a good book).Let M and N be crystal graphs, that is, coloreddirected graphs with the root operators ei and fi corre-sponding to movements along edges of color i. If v 2Mor N let "i(v) and �i(v) be the number of times ei or fimay be applied. As a set, M N is the Cartesian pro-duct, andfi(x y)=� fi(x) y if �i(x)>ei(y);x fi(y) if �i(x)6 ei(y);ei(x y)=� x ei(y) if �i(x)<ei(y);ei(x) y if �i(x)> ei(y):

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Creating all crystals� One method of generating all crystals: describeatomic ones and decompose tensor product (byextracting connected components of the tensorproduct graph.� For Ar you only need one atom, the standardmodule: � .-f1e1 � .-f2e2 � .-f3e3 �� By Frobenius-Schur duality, every irreducibleoccurs in some tensor power of the standard one.� Similarly for other classical groups you need onlya couple of atoms, but you need spin modules.� Identify the highest weight vector in a connectedcomponent of the tensor product by countinghow many total ei are needed to go from lowestweight to highest weight.

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Algorithms for Rank Two� A completely di�erent method that works wellfor rank 2 (may be tricky for G2 though the rele-vant formulas are known) is implemented in theC programs linked from wiki for A2 and B2.� The latter method: in the Littelmann cone model(where the vertices are string data), one rootoperator is obvious. This depends on a decompo-sition of the long element. Another decomposi-tion makes another root operator obvious. Bijec-tions between the two cones were given by Beren-stein and Zelevinsky (loc. cit.) and Littelmann(loc. cit.).� The C programs have the merit of drawing thevertices contributing to the same weight closetogether.

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Recommended Algorithms1. For Lie groups support two notations, here calledBourbaki notation and partition notation.2. Better expose Littelwood-Richardson rule fromsymmetrica.3. Highest weight modules are elements of subclassof rings or Z-algebras. Advantage: dimension andbranching rules are homomorphisms.4. Implement Weyl dimension formula as a homo-morphism to Z.5. Implement Freudenthal multiplicity formula.6. Have code to draw rank 2 weight diagrams.7. Implement one branch rule (e.g. Ar! Ar�1) as atemplate. Others can be added later at leisure.8. Brauer-Klimyk algorithm for tensor products.9. Crystal graphs should be implemented as coloreddirected graphs with access to root operators.10. Tensor product algorithm should be imple-mented. Computation of all crystals can bereduced to this.11. One graphical mode of representation should tryto keep vertices with same weight close together.28