IAP 2014: DIRECTED READING PROGRAM Classes

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.
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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
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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.
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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.

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