Geometric Representation Theory (Lecture 5)

October 22, 2007

Geometric Representation Theory (Lecture 5)

Posted by John Baez

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Felix Klein had a great idea: a lot of geometry is secretly group theory. Say you’ve got a group $G$ of symmetries, and it acts transitively on a set $X$ of geometrical figures of some type. This means that

$X \cong G/H$

for some subgroup $H \subseteq G$ , namely the ‘stabilizer’ subgroup — the subgroup that preserves a figure. So, you get types of figures from subgroups of your symmetry group.

But, there’s a lot more. Say you have some relation between figures of type $X$ and figures of type $Y$ that’s invariant under your symmetry group. For example: ‘a point lies on a line’.

This means you have a subset $R \subseteq X \times Y$ that’s invariant under the action of $G$ . You can think of this as an $X \times Y$ -shaped matrix of 1’s and 0’s: 1’s where the relation holds, 0’s where it doesn’t. But, such a matrix can be reinterpreted as a linear operator

$R: \mathbb{C}^X \to \mathbb{C}^Y$

The invariance condition then means this is an intertwining operator between permutation representations of $G$ .

A wonderful fact — though the proof is easy — is that we can get a basis of intertwining operators this way, called ‘Hecke operators’. We get this basis from ‘atomic’ invariant relations, meaning those that can’t be chopped up into a disjunction — a logical ‘or’ — of smaller relations. Another way to think about it: these atomic invariant relations are just $G$ -orbits in $X \times Y$ .

Soon we’ll use all this to take permutation representations of groups and chop them into irreducible representations. So: we’ll turning the insight of Klein around, and use geometry to study group representations!

Today, in the 5th lecture of our Geometric Representation Theory seminar, James Dolan works through some easy examples of Hecke operators.

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Lecture 5 (Oct. 11) - James Dolan on Hecke operators. Examples of the big theorem from Lecture 3: for any finite group GG and finite GG-sets XX and YY, there’s a basis of intertwining operators from ℂ X\mathbb{C}^X to ℂ Y\mathbb{C}^Y coming from GG-orbits in X×YX \times Y. These intertwining operators are examples of “Hecke operators”, and when X=YX = Y they span an algebra, called a “Hecke algebra”. GG-orbits in X×YX \times Y are also “atomic geometrico-logical relations between types of geometric figures”.

Example 1: GG is the isometry group of a cube. XX is the set of corners of the cube. YY is the set of edges.

Example 2: G=GL(4,F)G = GL(4,F) for some field FF. XX is the set of lines in projective 3-space, that is, DD-flags where DD is this Young diagram:

YY is the set of “complete flags”, that is, EE-flags where EE is this Young diagram:
Example Young Tableaux in SVG
- Streaming video in QuickTime format; the URL is
  http://mainstream.ucr.edu/baez_10_11_stream.mov
- Downloadable video (642 megabytes)
- Lecture notes by Alex Hoffnung
- Lecture notes by Apoorva Khare
- Lecture notes by Chris Rogers

Posted at October 22, 2007 10:24 PM UTC

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Re: Geometric Representation Theory (Lecture 5)

Let me know which lecture notes you like best. Alex Hoffnung’s are the cleanest-looking, and perhaps stick most closely to what’s written on the board. Apoorva Khare and Chris Rogers include more stuff that’s said but not written.

They’re all best as a supplement rather than a substitute for the videos. When Derek Wise was taking notes, I was very careful to write a lot of details on the whiteboard. Now that he’s gone, and we’re making videos, I’m explaining more in spoken words, and writing less… and Jim goes even further in this direction.

Posted by: John Baez on October 23, 2007 2:44 AM | Permalink | Reply to this

Hopf algebra dimension, functor enumeration

As to groups of symmetries, and functors between them, I like the enumerations of the abelian cases in this form:

A120733
Number of matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n.

n a(n)
0 1
1 1
2 5
3 33
4 281
5 2961
6 37277
7 546193
8 9132865
9 171634161
10 3581539973
11 82171451025
12 2055919433081
13 55710251353953
14 1625385528173693
COMMENT
Partial sums give A007322.

Dimensions of the graded components of the Hopf algebra MQSym (Matrix quasi-symmetric funcions). - Jean-Yves Thibon (jyt(AT)univ-mlv.fr), Oct 23 2006

LINKS
G. Duchamp, F. Hivert, and J.-Y. Thibon,Noncommutative symmetric functions
VI: Free quasi-symmetric functions and related algebras, Internat. J. Alg.
Comp. 12 (2002), 671-717
FORMULA
a(n) = (1/n!)*Sum_{k=1..n} (-1)^(n-k)*Stirling1(n,k)*A000670(k)^2. G.f.:
Sum_{m>=0,n>=0} Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*((1-x)^(-j)-1)^m.

a(n) = Sum_{r>=0,s>=0} binomial(r*s+n-1,n)/2^(r+s+2).

G.f.: Sum_{n>=0} 1/(2-(1-x)^(-n))/2^(n+1). - Vladeta Jovovic
(vladeta(AT)Eunet.yu), Oct 30 2006
CROSSREFS
Cf. A101370, A007322.
Cf. A120732.
KEYWORD
nonn
AUTHOR
Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 18 2006, Aug 21 2006

and

A007322
Number of functors of degree n from free abelian groups to abelian groups.

n a(n)
1 1
2 6
3 39
4 320
5 3281
6 40558
7 586751
8 9719616
9 181353777
10 3762893750
11 85934344775
12 2141853777856
13 57852105131809
14 1683237633305502
15 52483648929669119
16 1745835287515739328
17 61712106494672572641
18 2309989101145068446502
REFERENCES

H. J. Baues, Quadratic functors and metastable homotopy, Jnl. Pure and Applied Algebra, 91 (1994), 49-107.

D. Zagier, personal communication.
FORMULA

Binomial transform of A101370. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug
17 2006

a(n) = (1/n!)*Sum_{k=1..n} (-1)^(n-k)*Stirling1(n+1,k+1)*A000670(k)^2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 17 2006

G.f.: (1/(1-x))*Sum_{m>0,n>0} Sum_{j=1..n}
(-1)^(n-j)*binomial(n,j)*((1-x)^(-j)-1)^m. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 17 2006

Partial sums of A120733. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 21 2006
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
njas, Mira Bernstein (mira(AT)math.berkeley.edu)

EXTENSIONS
More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 17 2006

Now, what are the equivalent enumerations for nonabelian groups? For 2-groups? How does category theory and n-category theory assist the unenlightened such as myself in arriving at such enumerations?

Posted by: Jonathan Vos Post on October 27, 2007 11:03 PM | Permalink | Reply to this

Read the post Geometric Representation Theory (Lecture 7)
Weblog: The n-Category Café
Excerpt: James Dolan on two applications of Hecke operators: showing that any doubly transitive permutation representation is the direct sum of two irreducible representations, and getting ahold of the irreducible representations of $n!$.
Tracked: November 4, 2007 4:28 PM

Read the post Geometric Representation Theory (Lecture 11)
Weblog: The n-Category Café
Excerpt: How to describe Hecke operators between flag representations using matrices.
Tracked: November 16, 2007 9:04 PM

The n-Category Café

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October 22, 2007