Goals of this document This document tries to explain a few algorithms from Lie theory that would be useful to implement. I tried to include explanati

Transcription

1 Representation Theory Issues For Sage Days 7

2 Goals of this document This document tries to explain a few algorithms from Lie theory that would be useful to implement. I tried to include explanations of everything. There is too much to cover in my talk but I hope that the reference will be useful. A summary of recommended algorithms may be found at the end.

3 Modules for Lie groups, algebras How to represent modules of a Lie group or algebra to a computer? Use Cartan classication: A r (); B r (); C r (); D r (); E r (); F (); G () where is highest weight vector (HWV) HWV itself has alternative representations. Ring structure is important: should be able to add or multiply: A (; 0) A (0; ) = A (; ) + A (0; 0) gives decomposition of tensor product for SL 3. Ring structure makes dimension a homomorphism: Dim(A (; )) should produce 8. Branching rules are also homomorphisms. Therefore virtual modules should be a subclass of rings or Z-algebras. 3

4 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 Z r R X (T ) = t is a Euclidean vector space It is given a partial order. Positive elements are called dominant and form a cone, the positive Weyl chamber C +. Example: in A below and are simple positive roots, " and " are fundamental weights. The " i are dominant, i not. α ρ C + α

5 Example: SL3 = SU(3) = A The nite-dimensional representations of SL 3 (C) are same as maximal compact subgroup SU(3). Finite-dim'l representations of GL 3 (C) are the same as U (3). The representations of SL 3 or SU(3) are almost the same as GL 3 or U (3). G = SL 3, T = ft = diag(t ; t ; t 3 )jt t t 3 = g: X (T ) has basis with " (t) = t, " (t) = t 3. Embed X (T ) R 3 by ( ; ; 3 ) R 3 t t t t 3 3 : For SL 3, (; ; ) trivial character, so: X (T ) R 3 /R(; ; ) or f( ; ; 3 )j P i = 0g " (; 0; 0) or " (; ; 0) or 3 ; 3 ; 3 3 ; 3 ; 3 = (; ; 0) and = (0; ; ) are the simple roots. (Picture above.) 5

6 Highest Weight Vectors The Weyl group W acts on weights. The positive Weyl chamber is a fundamental domain for the action of W. Restricting the character of a representation to T it decomposes into a sum of weights, with multiplicities m() The function m() is invariant under W so its support intersects C +. Theorem. (Weyl) Every weight in C + is the highest weight vector (in the partial order) of a unique irreducible representation. Thus the representation should be parametrized by the highest weight vector: V (). 6

7 A Computer Notation Assume G is semisimple of rank r The fundamental weights " ; ; " r are a basis of the weights (assuming G is simply-connected) that are dominant. A special vector: is the sum of the fundamental weights. It is also half the sum of the positive roots. Use the standard labeling of " ; ; " r in the appendix to Bourbaki, Groupes et Algebres de Lie Ch IV,V and VI. Represent G by its Cartan classication. Computer notation: G(n ; ; n r ) is the irreducible module of G with HWV P n i " i. Thus B3[0,0,] is the spin module of so(7). I'll call this Bourbaki notation. Another notation is extensively used in the literature in which highest weight vectors are decreasing integer sequences. This notation is explained well in Goodman and Wallach, Representations and Invariants of the Classical Groups. We will call the notation Partition notation. We will emphasize it only for GL n but it also applies to other Lie groups as a good way to express branching rules. 7

8 Partition Notation For many purposes it is sucient to consider semisimple Lie groups. But GL r+ (or same: U (r + )) are reductive but not semisimple. Although they can be avoided it is better not to. G = GL r+, T = ft = diag(t ; ; t r+ )g diagonal torus. X (T Z r+. In this identication Z r character t t t r r. Fundamental weights: " i = (; ; ; 0; ; 0), i 6 r * i-th position A dominant weight > > > r : If also r > 0 then is a partition (of length 6 r). We'll call this partition notation even though if r negative is not a partition. Partition notation is useful for branching rules. Use partition notation for relationship between GL r+ and S k (Frobenius-Schur duality) Partition notation is used in Symmetrica. is 8

9 Littlewood-Richardson Rule The Littlewood-Richardson rule is implemented in Symmetrica as the Schur outer product. L It manifests itself as a homogeneous graded ring R = R k where R k is the free abelian group on the partitions of k and the multiplication R k R l R k+l is the Littlewood- Richardson rule (LRR) = X?k+l c The coecients c are nonnegative and may be dened combinatorially. LRR has various meanings. It is the tensor product rule for representations of GL r+. Branching rule GL(m + n) GL(m) GL(n). Induction rule S k S l S k+l. Structure constants for ring of symmetric polynomials with Schur polynomial basis. Structure constants for cohomology of Grassmannians with Schubert cocycle basis. 9

10 Branching Rules It seems important to implement branching rules for various inclusions of Lie groups. These are surveyed in: Howe, Eng-Chye and Willenbring (005). King, J. Phys. A 8 (975), 99. The most important are in the book of Goodman and Wallach. Some are simple to describe, some more complex. If a sage framework for treating these could be arrived at for treating them by implementing an easy one or two then harder ones could be implemented at leisure. Two candidates: The branching rule GL n GL n is particularly simple and important. The branching rule GL k+l GL k GL l is the Littlewood-Richardson rule and could be implemented by exposing symmetrica. Branching rule (GL n GL n ). If = ( > > n) and = ( > n ) are highest > > weight vectors then the irreducible module VGLn() of GL n restricted to GL n contains VGLn (n) if and only if and interleave; if so it occurs with multiplicity one. Interleaving means > > > > n. 0

11 Frobenius-Schur Duality There is a dictionary between the representations of GL n and S k. Both classes are indexed by partitions. A partition of k into 6 n parts indexes either: An irreducible module S k () of S k; or An irreducible module GL n () of GL n(c). The bijection: if V = C n then both groups act on k V and as a bimodule it decomposes k V = M?k S k () GLn (): LRR as tensor product formula If ; are partitions of length 6 n then M GL n () GLn () = LRR as branching formula GL n+m ()j GLnGLm = M ; c c GL n (): GL n() GL m() LRR for symmetric groups M Sk+l Ind S ( ksl S k () Sl ()) =?k+l If?k;?l: c S k+l ():

12 Dimensions, weight multiplicities Let G be a (simply-connected) semisimple Lie group, R X (T ) the weight lattice. It is divided into positive roots + and negative ones. If is a dominant weight the Weyl dimension formula gives a fast way of computing the dimension of the highest weight representation V (): dim V () = Y + h; + i ; = h; i X + : If the modules are implemented as a ring, this is a ring homomorphism Z. Mathematica: In[]:= < < lie.m In[]:= Dim[B3[0,,0]] Out[]= In[3]:= Dim[B3[0,0,]] Out[3]= 8

13 Tensor products Branching rules are also ring homomorphisms. We'd like the (virtual) modules over a reductive Lie group or algebra to be implemented as a Z- algebra so these homomorphisms can be handled as such. Call this Z-algebra K(G) or K(g). If SL 3 = A we'd like A(,0)*A(0,) to return A(,)+A(0,0) This can be implemented by better exposing the symmetrica function outerproduct_schur which implements the Littlewood-Richardson rule. This requires changing Bourbaki to partition notation. A(,0) becomes (,) A(0,) becomes (,) sage: symmetrica.outerproduct_schur((,),(,)) s[,, ] + s[, ] These are elements the ring of symmetric polynomials. There is a homomorphism to K(A ) in which s[,, ] + s[, ]>A (0; 0) + A (; ): Taking dimensions should give

14 Other Lie groups For other Lie algebras, taking the tensor products of representations can be done eciently using a method of Brauer and Klimyk. This requires computation of the weight multiplicities of one of the two representations. If is a highest weight vector, restrict the highest weight module V () to T and ask for the decomposition: V ()j T = X m() e : Here e is a synonym for intended to make the notation unambiguous. The coecients m() are essentially given by the Weyl character formula but are better computed algorithmically using Freudenthal's multiplicity formula. Freudenthal's formula: Humphreys, Introduction to Lie Algebras and Representation Theory Section.3. Brauer-Klimyk algorithm: Humphreys, Exercise 9 on page. Mathematica code implementing Freudenthals' formula for A, B and G :

15 Freudenthal Multiplicity Formula Let m() = m(; ) be the multiplicity of in highest weight module for. Recursively: [h + ; + i h + ; + i]m() = X X + i= 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() = and m is stable on W, so compute one value and you've computed up to jw j values. 5

16 Brauer-Klimyk method for We compute V () V (). The method is asymmetric. Decompose V () = m(; ). Suppose rst is so large that + C + for all with m(; ) 0. Then V () V () = M m(; )V ( + ): In the general case, discard such that + + is on a wall of a Weyl chamber. For the survivors nd w W such that w ( + + ) C +. Thene w ( + + ) is in the interior of C + and we may write w ( + + ) = 0 + where 0 C +. Now V () V () = M ( ) l(w ) m(; )V ( + 0 ): Since algorithm is asymmetric, choose to be the higher weight. 6

17 Drawing Rank Weight Diagrams Rank : R so display the weights and multiplicities in a diagram. A G weight diagram An A weight diagram A B weight diagram 7

18 Crystal Graphs One may replace the Lie group G with Lie algebra g by its quantized universal enveloping algebra U (g) which admits a deformation U q (g). This is a Hopf algebra whose modules are the same as those of G but the tensor product structure is dierent. If q it reverts to U (g) but if q 0 then a structure emerges known as the crystal graph. The vertices of the crystal graph correspond to vectors in a highest weight module. Let V be this module and let t be a weight. Then the weight space V () is the -eigenspace for t. If is a root of g and if X g is the eigenspace then X is one dimensional and a basis vector X maps V () V ( + ). If is a simple root then we may denote the operator X by e and X by f. Then e (resp. f ) permute the weight spaces, mapping V () 0 if + (resp. ) is not a weight. It is not possible to choose a basis of V that is simply permuted by the e and f. However by passing to U q (g) and (roughly) letting q 0 such a basis exists (Kashiwara). Every element of the crystal basis is mapped to another or to 0 by e and f. 8

19 A sample crystal graph Weight diagram for A (3; ), and the crystal graph: It is good to draw vertices contributing to the same weight space close together. This is possible for rank and sometimes rank 3. e = e and f shift right and left along green. e = e and f shift southeast and northwest along red. 9

20 Parametrizing the Crystal Basis There are several methods of parametrizing the crystal basis. For A r Gelfand-Tsetlin patterns; For B r, C r or D r modied half Gelfand-Tsetlin patterns; For A r or with modication B r, C r or D r semistandard Young tableaux (SSYT) In general, string representations (Berenstein- Zelevinsky, Littelmann). Another scheme due to Lusztig. Historically combinatorial literature emphasizes SSYT's for everything starting with the representation theory of S k but a trend may exist toward (equivalent) Gelfand-Tsetlin patterns. Alternating sign matrices are a special case. 0

21 String representations Berenstein and Zelevinsky, Canonical bases for the quantum group of type A r and piecewiselinear combinatorics, Duke (996). Littelmann, Cones, Crystals and Patterns, Transformation Groups (998). Pick a decomposition of the long element of the Weyl group into a product of simple reections. This will produce an embedding of the crystal graph into a cone in Euclidean space. There are many decompositions if r > 3 but Littelmann showed that one choice leads to a cone that is describable by explicit inequalities. Deformations of the Weyl character formula Deformations of the Weyl character formula express the modied numerator as a sum over Gelfand-Tsetlin patterns, SSYT's, Alternating sign matrices, etc. Older work of Tokuyama, Okada, Hamel and King and Simpson; More recent work of Brubaker, Bump, Chinta, Friedberg and Gunnells, a biproduct of investigations motivated from automorphic forms. These deformed Weyl character formulae may be expressed as sums over the crystal graph and the most important data are the string data.

22 Example: For A, w 0 = s s s. Read this as a word telling in what order to apply the lowering operators f i. Example: Green, Red, Green. Pick a vertex. To obtain the string, apply the green operator as many times x as possible; then red as many times y as as possible; then green as many times z as possible. At the end, you are at the unique lowest weight vector. string data = z y x This is not the Gelfand-Tsetlin pattern. The rows increase but do not interleave. This data uniquely determines the vertex and is a particularly useful way of representing it.

23 Gelfand-Tsetlin patterns Gelfand-Tsetlin patterns are a reection of multiplicity free branching rules. For type A, the multiplicity-free branching rule goes GL n GL n. The idea is that we can pick out an individual vector by asking that it lie in an irreducible subspace for each branching. In view of the branching rule described earlier, we need to specify a partition (or decreasing integer sequence) for each layer and these must interleave. The resulting pattern looks like this: 3 r r The SSYT is obtained by lling each skew partition etc. with an integer. 8 >< >: >= >; 3 3 3

24 String Data and GT patterns For the classical group, the connection between string data and Gelfand-Tsetlin patterns or tableaux was made explicit by Littelmann for a good decomposition of the long element. Accumulate the row dierences in the GT pattern adding the rst several elements of one row and subtracting the corresponding elements of the row below. Gelfand-Tsetlin= 8 >< >: Thus adding the red numbers and subtracting the green gives 6 which again goes in the blue spot in the array. Summary String Data= 8 >< >: For any g string data parametrize of the vertices of the crystal graph. They are also important in applications. For classical groups there are bijections 9 >= string data GT patterns tableaux >; 9 >= >;

25 Tensor products It turns out that the tensor product of two crystals has a very simple algorithm. Kashiwara and Nakashima, Journal of Algebra 65 (99). A very popular algorithm explained elsewhere, e.g. Hong and Kang, Introduction to Quantum Groups and Crystal Graphs (a good book). Let M and N be crystal graphs, that is, colored directed graphs with the root operators e i and f i corresponding to movements along edges of color i. If v M or N let " i (v) and i (v) be the number of times e i or f i may be applied. As a set, M N is the Cartesian product, and f i (x y) = fi (x) y if i (x) > e i (y); x f i (y) if i (x)6e i (y); e i (x y) = x ei (y) if i (x) < e i (y); e i (x) y if i (x)>e i (y): 5

26 Creating all crystals One method of generating all crystals: describe atomic ones and decompose tensor product (by extracting connected components of the tensor product graph. For A r you only need one atom, the standard module: e 3. - f e e f f 3 By Frobenius-Schur duality, every irreducible occurs in some tensor power of the standard one. Similarly for other classical groups you need only a couple of atoms, but you need spin modules. Identify the highest weight vector in a connected component of the tensor product by counting how many total e i are needed to go from lowest weight to highest weight. 6

27 Algorithms for Rank Two A completely dierent method that works well for rank (may be tricky for G though the relevant formulas are known) is implemented in the C programs linked from wiki for A and B. The latter method: in the Littelmann cone model (where the vertices are string data), one root operator is obvious. This depends on a decomposition of the long element. Another decomposition makes another root operator obvious. Bijections between the two cones were given by Berenstein and Zelevinsky (loc. cit.) and Littelmann (loc. cit.). The C programs have the merit of drawing the vertices contributing to the same weight close together. 7

28 Recommended Algorithms. For Lie groups support two notations, here called Bourbaki notation and partition notation.. Better expose Littelwood-Richardson rule from symmetrica. 3. Highest weight modules are elements of subclass of rings or Z-algebras. Advantage: dimension and branching rules are homomorphisms.. Implement Weyl dimension formula as a homomorphism to Z. 5. Implement Freudenthal multiplicity formula. 6. Have code to draw rank weight diagrams. 7. Implement one branch rule (e.g. A r! A r ) as a template. Others can be added later at leisure. 8. Brauer-Klimyk algorithm for tensor products. 9. Crystal graphs should be implemented as colored directed graphs with access to root operators. 0. Tensor product algorithm should be implemented. Computation of all crystals can be reduced to this.. One graphical mode of representation should try to keep vertices with same weight close together. 8