IAP 2014: DIRECTED READING PROGRAM REPRESENTATION THEORY OF FINITE GROUPS AND LIE ALGEBRAS Classes: WFS 12:00-14:00, Room E17-306 Instructor: Oleksandr Tsymbaliuk E-mail: sasha− [email protected] This intense course will be devoted to the basic aspects of the representation theory of finite groups and Lie algebras. In the first half of the course we will study the classical results on the representation theory of finite groups, including the character theory. We will conclude this discussion by classifying all irreducible representations of the symmetric group Sn and the alternating group An , and by discussing the Schur-Weyl duality. As an optional material, we will also derive the character table for the groups GL2 (Fq ) and SL2 (Fq ). In the second part of the course, we will go over the basic notions of a Lie group and a Lie algebra. We will discuss classical results on Lie algebras, such as Engel’s and Lie’s theorems, the Campbell-Hausdorff formula, and the Jordan decomposition. We will conclude the class by a classification of complex simple Lie algebras and the Weyl character formula. There are no official requirements to pass the course. However, it is highly recommended to do homework, which will consist of particular exercises from Fulton-Harris [1] and will be posted on the course website. There will be also a final home exam. The literature for the course is: [1] W. Fulton and J. Harris, Representation Theory: A First Course, book. [2] P. Etingof et al., Introduction to representation theory, arXiv/0901.0827. 1 2 REPRESENTATION THEORY OF FINITE GROUPS AND LIE ALGEBRAS Homework assignments The problems are borrowed from [1]. Therefore, we will only indicate their numbers. • Lecture 1: Representations of finite groups. Exercises: 1.3, 1.4, 1.12, 1.13, 1.14. • Lecture 2: Characters. Exercises: 2.3, 2.4, 2.7, 2.25, 2.27, 2.29, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39. • Lecture 3: Induced representations and group algebras. Exercises: 3.2, 3.4, 3.8, 3.11, 3.16, 3.19, 3.26, 3.38, 3.39, 3.40, 3.42, 3.43, 3.44. • Lecture 4: Representations of Sn . Exercises: 4.4, 4.6, 4.13, 4.16, 4.19, 4.20, 4.24, 4.40, 4.43, 4.44, 4.47, 4.51, 4.52. • Lecture 5.1: Representations of An . Exercises: 5.2, 5.4, 5.5. • Lecture 5.2: The character table of GL2 (Fq ) and SL2 (Fq ) (optional). Exercises: 5.7, 5.9, 5.10. Additional homework: work out all calculations from this section. • Lecture 6: Schur-Weyl duality and Schur functors. Exercises: 6.4, 6.10, 6.11, 6.13, 6.16, 6.17(a), 6.18, 6.21, 6.29, 6.30, 6.31. • Lecture 7: Lie groups. Exercises: 7.1, 7.2, 7.3, 7.6, 7.8, 7.11, 7.13, 7.14, 7.16, 7.17. • Lecture 8: Lie algebras and Lie groups. Exercises: 8.1, 8.10, 8.17, 8.24, 8.28, 8.29, 8.35, 8.36, 8.40, 8.43, 8.44. Additional homework: work out details in the proof of the Campbell-Hausdorff formula. • Lecture 9: Initial classification of Lie algebras. Exercises: 9.1, 9.2, 9.5, 9.7, 9.8, 9.10, 9.21, 9.22, 9.24. • Lecture 11: Representation theory of sl2 . Exercises: 11.11, 11.14, 11.17, 11.19, 11.23, 11.25, 11.33, 11.36. • Lecture 14: General structure theory of semisimple Lie algebras. Exercises: 14.14, 14.28, 14.33, 14.34, 14.35, 14.36. • Lecture 21: Classification of complex simple Lie algebras. Exercises: 21.6, 21.8, 21.15, 21.17, 21.18. • Lecture 24: Weyl character formula. Exercises: 24.4, 24.7, 24.20, 24.23, 24.27, 24.31, 24.43, 24.49, 24.50. • Appendix C: On complete irreducibility and Jordan decomposition. Exercises: C.1, C.13, C.14, C.28. • Appendix D: On the Cartan subalgebras and the Weyl group. Exercises: D.5, D.24, D.30, D.35, D.38, D.41. • Appendix E: Ado’s and Levi’s theorems (optional). [1] W. Fulton and J. Harris, Representation Theory: A First Course. IAP 2014: DIRECTED READING PROGRAM 3 Home Exam Part I 1. Let G be the group of nonconstant linear transformations x 7→ ax + b over Fq . (a) Find all irreducible G-representations and compute their characters. (b) Compute the tensor product of irreducible representations. 2. Let X1 , X2 be two G-sets, so that C[X1 ] and C[X2 ] become G-representations. (a) Find c(X1 , X2 ) := dim HomG (C[X1 ], C[X2 ]). (b) Let X1 and X2 be homogeneous G-spaces, so that Xi = G/Hi . Show that C[Xi ] = IndG Hi C and prove that c(X1 , X2 ) = # H1 \G/H2 . (c) Let us consider the action of G×G on G given by (g1 , g2 )◦g := g1 gg2−1 , and let Reg be the corresponding G × G-representation on C[G]. Prove that dim HomG×G (Reg, Reg) is equal to the number of G-conjugacy classes on the one hand, and to the number of irreducible finite dimensional G-representations on the other hand. 3. Let G := GLn (Fq ) and Xkq := Grk (Fnq )–the space of k-dimensional subspaces of Fnq . The natural action of G on Xkq induces a G-action on C[Xkq ] =: Vkq . (a) Compute #G and #Xkq . q (b) Prove that G-representations Vkq and Vn−k are isomorphic. q q (c) Prove that dim HomG (Vk , Vl ) = 1 + min{k, l, n − k, n − l}. ⊕min{k,n−k} q q (d) Prove that Vkq = i=0 Ui for some G-irreducible representations U0q , . . . , U[n/2] . Part II 4. (a) Decompose IndSS43 π into irreducibles for every irreducible S3 -representation π. (b) Decompose IndSS53 ×S2 sgn ⊗ sgn into irreducibles. (c) Decompose ResSS42 ×S2 π into irreducibles for every irreducible S4 -representation π. 5. Consider a subgroup Zn ⊂ Sn generated by the long cycle σ = (12 . . . n), and a character χ : Zn → C∗ with χ(σ) being a primitive n-th root of 1. (a) Decompose IndSZnn χ into irreducibles for n = 3, 4. (b) Find the multiplicities of V(1n ) = sgn and the standard representation V(n−1,1) in IndSZnn χ. (c) In general, show that the multiplicity of Vλ in IndSZnn χ is given by the following formula: 1∑ µ(d)χVλ (σ n/d ), n d|n where µ(d) is the M¨obius function. 6. Let us define an element Cn := ∑ i<j (ij) ∈ CSn . ∑ ∑ λj (a) Show that Cn acts on Vλ as a multiplication by the scalar cλ = j i=1 (i − j). (b) Show that En := (12)+. . .+(1n) ∈ CSn acts diagonalizably on Vλ with integer eigenvalues from {1 − n, 2 − n, . . . , n − 2, n − 1}. (c) Show that En acts on Vλ as a multiplication by a scalar iff λ is a rectangular Young diagram. Compute this scalar in the latter case. 4 REPRESENTATION THEORY OF FINITE GROUPS AND LIE ALGEBRAS 7. Recall that ResSAnn Vλ is irreducible iff λ ̸= λ∗ . If λ = λ∗ it decomposes into a sum of two conjugate An -irreducibles: ResSAnn Vλ = Vλ′ ⊕ Vλ′′ . Compute the characters of Vλ′ and Vλ′′ . Hint: See [Fulton-Harris, Exercise 5.4] for the outline of key steps. 8. As we know, the group G = SL2 (F3 ) has 7 irreducible representations {Vi }7i=1 . Let V7 GL (F ) denote the representation ResSL22(F33) Xφ in the notation of [Fulton-Harris, p. 70] (it is also recommended to check that this restriction is indeed irreducible and does not depend on φ). Draw the graph, whose vertices are parametrized by {1, . . . , 7} and the number of edges between vertices #i and #j is equal to dim HomG (Vj , Vi ⊗ V7 ). 9. Prove that PSL2 (Fq ) is a simple group for odd q > 3. Is PSL2 (F3 ) simple? Part III 10. Classify irreducible finite dimensional representations of the two-dimensional Lie algebra with basis X, Y and commutation relation [X, Y ] = Y . Consider the cases of zero and positive characteristic (we work only over algebraically closed fields). 11. Let L be a free Lie algebra on k generators x1 , . . . , xk . Consider a grading on L with ⊕ deg(x1 ) = . . . = deg(xk ) = 1. Thus L = n≥0 Ln with Ln being the degree n component. Prove the following formula: 1∑ µ(d) · k n/d , dim Ln = n d|n where µ(d) is the M¨obius function. Hint: Use the M¨obius inversion formula. 12. Let g be a simple Lie algebra and h ⊂ g–its Cartan subalgebra. Let R = R− ∪ R+ be the set of all roots of g and Π ⊂ R+ be the∑set of positive∑ simple roots. Choose nonzero elements eα ∈ gα for each α ∈ Π and set e := α∈Π eα , h := α∈R+ hα . Show that there is a unique element f ∈ g such that {f, h, e} generate a subalgebra isomorphic to sl2 via F 7→ f, H 7→ h, E 7→ e.
© Copyright 2025 ExpyDoc