Lecture 1: Introduction to the summer material.
Symmetric functions, Lie algebras, and their representations
The goals for ICLUE Summer 2022.
* I wish to explain to you some basic concepts of higher algebra
(representation theory). My overriding philosophy is the same as in 347H:
to make sure you know all the basic things so as to not embarrass UIUC
in front of a reasonable person. Except now the standard of reasonable person
would be not a "New York city subway passenger" but potential future math mentors.
* Specifically, we will EVENTUALLY discuss the case of complex semisimple Lie
algebras.
* These subjects tie into many notions in pure mathematics (and in fact
physics, for real). To learn Lie theory is a core concept of
graduate/research level algebra, and won't be lost times.
* One of these is algebraic combinatorics (specifically symmetric functions), my
own field of research.
* The plan is to get to the point of understanding enough to do calculations,
study the literature, and learn on your own the details. There's much to do!
* As a consequence we may very well get to some novel research, but I need you to
build up enough understanding to be able to help out!
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How this differs from Math 347H -- A number of you have just taken Math 347H with
me. The basic plan is the same: definitions, MEANS, etc. However, were biting off
a lot more in this topic, and I do not pretend to know all the details. I'll be
(re)learning things as we go along. There may be things I simply don't know: you
can find out and lecture to us about it! Many questions one could ask might not
be known to anyone, or quickly touch on the horizon of understanding, or simply
be complicated enough that *I* cannot give easy answers.
I'll be needing YOU to help ME understand things, as well as vice versa.
Welcome to mathematical research!
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Groups: Before I get into Lie algebras, I need to explain a more undergraduate
concept, that of a group.
* Usually a course in group theory has a syllabus like Math 417 that sounds like:
"(a) The Integers Division algorithm. Greatest common divisor. Fundamental theorem
of arithmetic. Congruence arithmetic; application to RSA-cryptosystem. [4]
(b) Permutations Cycle decomposition. Order of a permutation. Even and odd
permutations. [3]
(c) Group Theory Definition and examples. Subgroups, cosets and Lagrange's
theorem. Normal subgroups and quotient groups. Homomorphisms. The Isomorphism
Theorems. [10]
(d) Group Actions Cayley's theorem. Burnside's theorem. Conjugacy classes and
centralizers. Applications of group actions, eg. to Sylow's theorem or Polya
counting. [10]
(e) Ring Theory I Definition and examples. Polynomial rings. Subrings, ideals and
quotient rings. Homomorphisms of rings. The Isomorphism Theorems for rings.
Integral domains and fields. Division algorithm for polynomial rings over a
field. Roots of polynomials and the Remainder Theorem. The Fundamental Theorem of
Algebra (without proof). Maximal ideals in polynomial rings over fields, with
application to the construction of fields. [12]"
https://math.illinois.edu/resources/department-resources/syllabus-math-417
I'm not teaching Math 417 here. We'll talk about things as they come up and sometimes
I'll take some things as facts which you can read up on your own.
I will focus on something that is NOT taught much in undergrad (usually) and
reserved for advanced classes in grad school, DESPITE that something being the
motivation for group theory to begin with!!
That something is REPRESENTATION THEORY.
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Definition: A *group* is a pair (G,*) where g is a set and *:G x G ->G
is a binary operation satisfying the following axioms:
Associativity: (a*b)*c = a*(b*c)
Identity: there exists e in G such that for all a in G, e*a=a=a*e
Inverse: For every a in G there exists b in G such that a*b=e=b*a, we write
b=a^{-1}.
ex. (Z,+)
ex. The free group on n letters. Say n=3: a,b,c. Product is concatenation, and
define a^{-1}, b^{-1}, c^{-1} to exist. Identity is the empty word.
The free group F(X) on a set X is *universal* in the following sense
F(X)->G
^ /
| /
| /
| /
X
Suppose f:X->G is any function, and X-->F(X) is the inclusion map.
Then there is a unique homomorphism Q:F(x)->G such that the diagram commutes.
The free group is a "functor from the category of sets to the category of groups".
*Category* C: has objects and morphisms Hom(X,Y) for all objects X,Y in C, with
composition having what you expect: associative, and there exists an identity
*Functor* is a mapping F between categories C and D
that
* sends objects to objects
* morphisms f:X->Y to morphisms F(g):F(X)->F(Y) such that
F(id_X)=id_{F(x)} F(g o f) = F(g) o F(f) for morphisms f:X->Y, g:Y->Z in C
ex. Free group quotient by relations F/I. Here I= and
quotient means quotient of a group by a normal subgroup I.
*Normal subgroup* means gig^{-1} in I for all i in I and g in G.
This is the correct notion of things to quotient by, as I now explain:
Recall that given some category of objects we have the notion of a homomorphism.
These are maps that "preserve the structure of the object":
Definition: A *homomorphism* of groups f:G->H is a function such that
f(a*_Gb)=f(a)*_H f(b).
Exer: It follows f(e_G)=e_H, f(a^{-1})=f(a)^{-1}.
Now suppose f:G->H is a homomorphism. Let Ker(f)={g in G: f(g)=id_H}.
Notice that Ker(f) is a subgroup (why?).
Notice moreover that Ker(f) is a NORMAL subgroup: Let n in Ker(f). Then
f(gng^{-1})=f(g)f(n)f(g^{-1})=f(g)id_H f(g)^{-1} = f(g)f(g)^{-1}=id_H
Hence gng^{-1} in Ker(f) as well.
In general in algebra, the correct subobjects to quotient by are precisely
those that have the property of Kernels of maps.
Quotient G/N: define an equivalence relation ~ on G by g~g' if g(g')^{-1} in N.
(Of course) groups have the 1st isomorphism theorem:
Suppose f:G->>H is a surjection, then H iso G/ker(f). (Exer.)
------------------
ex. C_n = Cyclic groups of order $n$: F/.
ex. Cyclic groups in nature: Z_p^* = {1,2,3,4,...,p-1} under modular product
(modulo p). (why?)
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ex. Dihedral groups: D_n=F/.
ex. Dihredal groups in nature: The symmetries of a regular n-gon (r=rotation, s=
reflection)
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ex. "Symmetries of a shape", e.g., symmetries of a cube.
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Two major players this summer are the next two examples:
ex. The symmetric group S_n.
- Explain S_n basics.
ex. The Lie group GL_n(C)
- Explain GL_n(C) basics. The determinant for example.
Definition: Let GL(V) be the set of invertible linear transformations of
a finite dimensional vector space V.
(It is equivalent to think GL(V)=GL(C^n) for n=dim(V).)
Definition: A *linear* representation of a group G is a homomorphism
f:G->GL(V) for some finite dimensional vector space V.
Definition: (Somewhat confusingly) the *degree* of a linear representation
is the dimension of V.
Fact to discuss: it is equivalent to think of V being an R-module where
R=C[G]=the group ring of G. (Explain.)
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Example: A dumb representation that's always available is the map f:G->GL(V) that
sends g|-> the identity matrix. This is called the *trivial* representation.
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Example: Let G=(Z,+) and V=R^3 with
T(n)=
[1 n (n^2+n)/2]
[0 1 n ]
[0 0 1 ]
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Example: Let G=complex numbers of absolute value 1 (product). This is
called the "circle group" U(1). Let V=C. For any integer n let
x|->z^n. These are representations of G. If instead we let V=R^2 then
we have representations that can be written as
f(e^{i theta})=
[ cos(n theta) -sin (n theta)]
[ sin(n theta) cos (n theta)]
and the trivial representation.
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Example: Let G=S_n and let f:S_n->GL_n(C) be the map that sends a
permutation to its permutation matrix.
Ex. w=31524 maps to
01000
00010
10000
00001
00100
If I did things right, this will be a homomorphism. (If I did it
wrong, I should use the transpose.)
This is called the *defining representation* of S_n.
--------
Example: Let sgn(w) be the sign of a permutation. This is
equivalently the determinant of its permutation matrix or the number
swaps one needs to go from the identity to w using *transpositions*.
The map f:S_n->C that sends w|->sgn(w) is also a representation.
--------
Example: GL(n) example. Let V=Sym^2(C^2) be the vector
space that is C-spanned by x^2, 2xy, y^2
Let GL_2 act on V by change of coordinates
x |->ax+cy
y |->bx+dy
Hence
x^2 |-> (ax+cy)^2, 2xy|-> 2(ax+cy)(bx+dy), y^2|->(bx+dy)^2
The associated homomorphism sends
[a b]
[c d]
to
[a^2 2ab b^2]
[ac bc+ad bd ]
[c^2 2cd d^2]
Exer: Check that if
[a b]
det[c d] \neq 0
then so is the det of
[a^2 2ab b^2]
[ac bc+ad bd ]
[c^2 2cd d^2]
Also check homomorphism property.
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Ex. A different sort of representation of GL(n)
[1 ln|det(g)|]
g|-> [0 1 ]
This is a non-rational representation of GL(n); no classification of such
representations exists.
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Example: (Group actions and permutation representations).
Given a group G and a set X one defines a (left) group action to be a
function Q:G x X ->X
satisfying two axioms
Q(e,x)=x
Q(g,Q(h,x))=Q(gh,x).
Usually in math we shorten the notation to
e.x=x
gh.x=x
Observation: Fix g, the map x|-> Q(g,x) is a bijection of X to itself. (Why?)
Now suppose X is finite. So x={a,b,c,d,...} We can create a vector space V
with basis B_a, B_b,B_c,... and let G act by sending B_a to B_{g.a}, etc.
This defines the *permutation representation* of G on V.
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Example: The regular representation of a finite group G. Let C[G] be
the group ring of G. One can think of it as a vector space whose
elements are linear combinations of group elements. Then R=C[G] acts
on itself: that is C[G] is a C[G] module. This is the regular
representation.
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Definition: If V is a vector space over a field k then a *linear
functional* is a linear map f:V->k. The vector space of linear
transformations from V to W is denoted Hom(V,W) and in particular,
the vector space of linear functionals f:V->k is denoted
V^*=Hom(V,k).
More on dual spaces and some exercises here:
https://mathinmoscow.org/wp-content/uploads/BRTS21_Lecture03.pdf
Here's a fact/definition we'll need
Suppose f in Hom(V,W) is a linear transformation. The *dual map*
f^* given by the transpose matrix is a map in Hom(W^*,V^*).
This is the map that sends a linear functional H in W^* to
H(f) in V^*.
(Read/exer.)
------------------------------------------------------------------
Example: Suppose r:G->GL(V) is a representation, then V^* is a
G-module under the action that sends f in V^*
by the map r^*:G->GL(V^*) to the linear functional r^*(g)
defined by
r^*(g)(f):= [r(g^{-1})^T](f)
The formula seems a bit weird! What's with the g^{-1}?
In fact the proof makes sense of it:
Theorem: r^*:G->GL(V^*) is a linear representation.
Proof: Well-defined: r(g^{-1}) in Hom(V,V). Hence (r(g^{-1}))^T is in
Hom(V^*,V^*). Also, if r(g^{-1}) in GL(V) then its transpose is
invertible. Hence (r(g^{-1}))^T in GL(V^*).
r^*:G->GL(V) is a group homomorphism:
Let g,h in G
r^*(gh):=r((gh)^{-1})^T=(r(h^{-1}g^{-1}))^T
(since (gh)^{-1}=h^{-1}g^{-1} for groups)
=(r(h^{-1}) r(g^{-1}))^T
(since r is a group homomorphism)
= r(g^{-1})^T r(h^{-1})^T
((AB)^T=B^T A^T for matrices)
= r^*(g) r^*(h) as desired.
QED
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Definition: (Restriction) Let H be a subgroup of G. Suppose
f:G->GL(V) is a linear representation. Then f|_H:G->GL(V) is
a linear representation of H.
Other constructions of representations from known ones: via tensor
product, via "induction". (We will talk about these a bit later.)
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DIRECT SUM
First some facts from Linear algebra:
Definition: Let V and W be vector spaces. The (external) *direct sum*
V \oplus W is the the vector space consisting of elements (v,w): v\in
V, W \in W.
Equivalent Definition: Suppose V is a vector space and W, W' are two
subspaces of V. Suppose W intersect W' = 0 and dim(V)=dim(W)+dim(W')
(equivalently, each x in V is UNIQUELY expressible as x=w+w' where w
in W and w' in W'). Then we say V=W\oplus W' (internal direct sum).
In this situation we say W' is the *complement* of W in V.
Exer: The two notions are compatible, the phrasing is different
depending on whether you fix the ambient vector space first and talk
about subobjects or fix the summands first.
CONNECTION TO PROJECTION
Definition: Let W be a subspace of V. A map f:V->V
such that
* f(x)=x if x in W
* im f =W
is called a *projection map* of V to W.
Exer: If f:V->W is a projection map then V=W \oplus ker(f) (internal
direct sum).
Exer: Every subspace W of V has a complement (use extension of
basis).
Exer: Thus, projections of V to subspace are in bijective
correspondence with complements.
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Definition: Now suppose V and W are G-modules associated to
homomorphisms f:G->GL(V) and g:G->GL(W). Then the *direct sum*
V\oplus W is the vector space where G acts by h(v,w)=(f(v),g(w)).
Theorem: V \oplus W is G representation if V and W are.
Proof: clear. QED
Less obvious is the following claim:
Theorem: Suppose G is a finite group and let f:G->GL(V) be a linear
representation. Suppose W is a vector subspace of V that is stable
under G (hence W is a G-subrepresentation), then there is a G-stable
complement W' of W in V. That is V decomposes as W\oplus W'.
Proof: Given W, find a complement W'. Unfortunately, W' might not be
G-stable, so we aren't done. However, let p:V->W be the projection
map associated to this complement. Look at the average p^0 of
conjugates of p
p^0=(1/|G|)\sum_{t in G} f(t).p.f(t^{-1})
Since p:V->W and f(t) preserves W (W is G-stable) we see p^0:V->W.
Let x in W, then
p.f(t)^{-1}x=f(t)^{-1}x (p is a projection of V->W)
f(t).p.f(t)^{-1}x=x
hence
p^0(x)=x.
Thus p^0 is a "better projection" of V->W. Let the complement be W^0.
Claim: f(s)p^0 = p^0 f(s)
Proof: Compute f(s).p^0 f(s)^{-1}
= 1/|G| \sum_{t\in G} f(s).f(t).p.f(t)^{-1}.f(s)^{-1}
= 1/|G| \sum_{t\in G} f(st).p.f(st)^{-1} = p^0. QED of claim
Finally, let x in W^0 and let s in G. Since p^0(x)=0, from the CLAIM
p^0.f(s)x=f(s).p^0x=0.
That is f(s)x in W^0 => W^0 is stable under G, which
is what we wanted. QED
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Definition: A G-representation V is *indecomposable* if V is not the
nontrivial direct sum of G-modules A and B.
Definition: A G-representation V is *irreducible* if it has no
nontrivial G-submodules. We say V is reducible otherwise.
We will work in the "semisimple" theory, which means
"indecomposible=irreducible".
That is, every reducible G-module V is a direct sum of irreducibles.
Remark: Our finite group proof above shows finite groups are
semisimple. If V is G-reducible (has a nontrivial subrepresentation
W) then V=W\oplus W' as G-representations.
Example: Let G=S_n and V=C^n. Let S_n act "naturally" on C^n by
permutation of the coordinates. Notice that V is not irreducible! The
subspace H defined by the hyperplane x1+x2+...+xn=0 is G-stable. The
complement of H is called the standard representation (and is in fact
irreducible (exer)
Proof idea: That complement W has basis e1-e2,e2-e3,...,e(n-1)-en.
Now argue that if v=a_1(e1-e2)+...+a_{n-1}(e_{n-1}-en) then the S_n
orbit of v contains all the basis vectors, which implies that any
S_n-subspace of is the entire thing. QED
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Example: Let us continue with U(1)=the circle group of complex
numbers of modulus 1. Let V be the space of 2\pi-periodic functions
(functions on a circle). Clearly U(1) acts on V, that is V is a
representation of U(1). Now decompose V into a direct sum of
U(1)-irreps. These irreps are one-dimensional. That is, any vector v
in V (a 2\pi-periodic function) is writable (uniquely) as a sum of
vectors (functions) on each of these irreps. This is called the
*Fourier series* of f.
----------------
Definition: Two G-modules V, W are *isomorphic* if there is a
G-equivariant homomorphism f:V->W that is bijective. By
G-equivariant, we mean f(g.v)=g.f(v).
Said another way, if f:G->GL(V) and g:G->GL(W) are two
representations, we consider them *equivalent* if there is an
isomorphism T:V->W of vector spaces such that
T f(group elt)=g(group elt) T
for all group elt in G.
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In class exer: Let G=(Z,+). Let us describe ALL representations of G. Let
f:Z->GL(m) be a representation. Then f(0)=Id. Given ANY invertible m
x m matrix M if we assign f(1)=M then f(n)=f(1+1+...+1)=M^n is
forced. Hence all representations of G are found this way.
But when are two such representations f:Z->GL(m) and g:Z->GL(m)
isomorphic/equivalent? Say the associated matrices are M, N.
By definition you want an invertible linear transformation T:C^m->C^m
(say defined by a matrix P) such that GT=TF or G=TFT^{-1}, that is G
and F are similar matrices. By linear algebra, every matrix is ~ to
a matrix in Jordan Canonical form (if we work over an algebraically
closed field). These are matrices of block form where each block
looks like
[a1000]
[0a100]
[00a10] =Jordan block.
[000a1]
[0000a]
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In class exer: Does the decomposition of a G-module into irreducibles
need to be unique?
Answer: No. Let G be any group and f:G->GL(V) be the trivial
representation. Then any positive dimensional subspace is stable
under the action of G. Thus if one chooses any line in V, it will be
stable, and its complement is stable. That is, any decomposition of V
into a direct sum of lines is a decomposition into irreducibles. But
there are (infinitely) many of these!
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Basic questions:
1) What are the irreducible representations? Can you combinatorially
name them all up to isomorphism?
2) What are the dimensions of these irreducible representations,
i.e., if f:G->GL(V) is an irreducible representation, what dim V's
are possible? Can you give a formula?
3) How do you construct the irreducibles? How do you construct
interesting representations (even if they are not irreducible).
4) Given an "interesting representation" how do you decompose it into
irreducibles. How many times does a copy of a given irreducible
appear in the decomposition? Can you give a combinatorial formula?
Books
Serre: for finite groups
http://www.math.tau.ac.il/~borovoi/courses/ReprFG/Hatzagot.pdf
Combinatorial Representation Theory
https://arxiv.org/pdf/math/9707221.pdf
Sagan's "The symmetric group" (findable online or legally through mathscinet)
Stanley's GL(n) for combinatorialists
https://math.mit.edu/~rstan/pubs/pubfiles/57.pdf
Youtube videos by Borcherds
Next: characters (finite groups), characters of GL_n.
Connections to symmetric function theory.
Frobenius map
Some slides
Open problems: https://www.maths.usyd.edu.au/u/geordie/Dusseldorf.pdf
https://mathoverflow.net/questions/96202/open-problems-questions-in-representation-theory-and-around