Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-in-frequency Fast Fourier Transforms
for the Symmetric Group
Eric Malm
Joint Mathematics Meetings, Atlanta ’05
Thesis Advisor: Michael Orrison
Harvey Mudd College
08 Jan 2005
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Overview
1 Group Rings and Wedderburn’s Theorem
Group Rings
Wedderburn’s Theorem
2 Discrete Fourier Transforms
Generalized DFTs
DFT as a Change of Basis
3 Fast Fourier Transforms for Sn
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
4 Future Work
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Group Rings
Wedderburn’s Theorem
Group Rings
We focus primarily on complex group rings CG: finite C-linear
combinations of elements of a group G.
Example (CS3 )
Let G = S3 ; then elements of CS3 are of the form
3 (1) + 47i (1 2) + (17 + 24i) (1 3 2)
with the group elements of S3 written in cycle notation.
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Group Rings
Wedderburn’s Theorem
Wedderburn’s Theorem
Theorem (Wedderburn)
The group ring CG of a finite group G is isomorphic to an algebra of
block diagonal matrices:
h
CG @
Å Cdj ´dj
j= 1
Example (CS3 )
CS3 @ C1´1 Å C2´2 Å C1´1
æçê
ö÷
çç
÷
çç ê ê ÷÷÷
÷÷
= ççç
çç ê ê ÷÷÷
ç
÷
êø
è
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Generalized DFTs
DFT as a Change of Basis
Generalized Discrete Fourier Transforms (DFTs)
Definition (Generalized DFT)
Any such isomorphism D on CG is a generalized DFT for G
• Coefficients in matrix D(f ): generalized Fourier coefficients
• Blocks along diagonal: smallest CG-invariant spaces in CG
Reduction to Classic DFT
When G = Z/ NZ, any such D reduces to the Discrete Fourier
Transform on N points
• DFT takes time-domain data into frequency spaces that are
invariant under translations
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Generalized DFTs
DFT as a Change of Basis
DFT as a Change of Basis
Change of Basis
h
d ´d
Fixing a D : CG ® Åj=1 C j j gives two standard C-bases for CG:
• “time-domain” basis S = {g}gÎG ,
• “frequency-domain” basis F = {D-1 (Ek,ij )},
where Ek,ij is the matrix with 1 in the ijth element of the kth block
and 0s elsewhere.
The DFT matrix is then the change-of-basis matrix from S to F.
Bound on Operation Counts
Formulation as a matrix operation gives naïve bound of O(|G|2 ) on
complexity for DFT evaluation
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Decimation-In-Frequency Fast Fourier Transforms (FFTs)
Decimation-In-Frequency FFT
• Fix chain of subgroups of G:
1 = G0 < G1 < × × × < Gn-1 < Gn = G.
• Project into successively smaller subspaces in stages
corresponding to subgroups
• Goal: Obtain sparse factorization of change-of-basis matrix D
Sn an Ideal Proof-of-Concept Group
Nonabelian, representation theory well understood, natural chain of
subgroups
1 < S2 < S3 < × × × < Sn
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Representation Theory of Sn
Each block in matrix algebra for CSn corresponds to a different
non-increasing (proper) partition of n
Example (CS3 )
CS3 @
IêM
ê ê
Å
Kê êO Å IêM
3#
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Bratteli Diagram for 1 < S2 < S3
Graded diagram of proper partitions for 1 < S2 < × × × < Sn
æçê
ö÷
çç
÷
çç ê ê ÷÷÷
çç
÷
çç ê ê ÷÷÷
çç
÷÷
êø
è
CS3
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Bratteli Diagram for 1 < S2 < S3
Each block for Sn restricts to a multiplicity-free direct sum of blocks
for Sn-1 by removing a box
æçê
ö÷
çç
÷
çç ê ê ÷÷÷
çç
÷
çç ê ê ÷÷÷
çç
÷÷
êø
è
CS3
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Bratteli Diagram for 1 < S2 < S3
Each pathway through the diagram corresponds to a filled-in partition
and a row/column in matrix
ö÷
æçê
çç
÷
çç ê ê ÷÷÷
÷
çç
çç ê ê ÷÷÷
÷÷
çç
êø
è
1 2 3
1 2
CS3
1
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Bratteli Diagram for 1 < S2 < S3
Each pathway through the diagram corresponds to a filled-in partition
and a row/column in matrix
1 3
2
ö÷
æçê
çç
÷
çç ê ê ÷÷÷
÷
çç
çç ê ê ÷÷÷
÷÷
çç
êø
è
1
2
CS3
1
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Bratteli Diagram for 1 < S2 < S3
A pair of paths ending at same diagram specifies a 1-D Fourier space
¯
æçê
ö÷
ç
÷
® ççç ê ê ÷÷÷
çç
÷
çç ê ê ÷÷÷
çç
÷÷
êø
è
CS3
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Construction of Factorization
Stages of Subspace Projections
• Partial paths give CSn subspaces
• At stage for Sk , project onto
these subspaces
• Build sparse factor from
projections
• Full paths by stage for Sn
CS3 @ I ê M Å K
ê ê
O Å IêM
ê ê
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Construction of Factorization
Stages of Subspace Projections
• Partial paths give CSn subspaces
• At stage for Sk , project onto
these subspaces
• Build sparse factor from
projections
• Full paths by stage for Sn
CS3 @ I ê M Å K
ê ê
O Å IêM
ê ê
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Factorization of CS3 DFT Matrix
Full DFT matrix for CS3 :
æç1 1
çç
çç1 -1
çç
çç
çç0 0
çç
çç
çç0 0
çç
çç
çç1 1
çç
ç
è1 -1
1
1
1
1
2
- 34
- 12
3
4
1
2
3
4
1 ö÷
÷÷
- 12 ÷÷÷÷
÷÷
- 34 ÷÷÷÷
÷÷
-1 -1 1
1 ÷÷÷
÷÷
- 21 - 12 - 21 - 12 ÷÷÷÷
÷÷
-1 1 -1 1 ø
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Factorization of CS3 DFT Matrix
Three factors (CS2 on rows, CS2 on columns, CS3 on rows):
æç11
çç
çç
çç
çç
çç
ç
è
1
- 21
1
1
1
1
ö÷ æç1
÷÷ çç
÷÷ çç
÷÷÷ ççç
÷ çç
1 ÷
֍
2 ֍
-1ø è
1
1
1
-1
1
1
1
ö÷ æç1
÷÷ çç
÷÷ çç
÷ çç
÷÷ çç1
-1 ÷
÷÷ çç
֍
1 øè
1
1
1
1
-1
1
-1
1
ö÷
÷÷
1÷
÷÷
÷÷
÷÷
÷÷
-1 ø
plus a permutation matrix and a row-scaling diagonal matrix
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Decimation-In-Frequency FFTs
Representation Theory of Sn
Construction of Factorization
Initial Results
Results from Initial Implementation
Initial þÿ Implementation
Can compute exact FFT up to n = 6
Operation Counts For Evaluation
n
Å
Ä tDIF tMaslen [1]
1
n(n
2
- 1)
3
14
3
2.5
2.7
3
4
112
43
5.0
5.4
6
5
986
419
8.8
9.1
10
6
9696
4634
14.2
13.6
15
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
Next Steps
Theory
• Prove O(n2 |Sn |) bounds on operation counts
• Deduce better bases for blocks in factors
• Relate FFT on Sn to FFTs on Sn-1
Experiments/Implementation
• Improve efficiency of þÿ implementation
• Run factors through SPIRAL system
• Port to Matlab or GAP
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group
Group Rings and Wedderburn’s Theorem
Discrete Fourier Transforms
Fast Fourier Transforms for Sn
Future Work
References
[1] David K. Maslen.
The efficient computation of fourier transforms on the symmetric group.
Mathematics of Computation, 67(223):1121–1147, July 1998.
[2] David K. Maslen and Daniel N. Rockmore.
Generalized FFTs - a survey of some recent results.
DIMACS Series in Discrete Mathematics and Computer Science, 28:183–237,
1997.
[3] Bruce E. Sagan.
The Symmetric Group: Representations, Combinatorial Algorithms, and
Symmetric Functions.
Wadsworth & Brooks/Cole, Pacific Grove, CA, 1991.
On The Web
Senior thesis website: Xhttp://www.math.hmc.edu/~emalm/thesis/\
a·r·t
har vey·mudd·college
Eric Malm
FFTs for the Symmetric Group