Adapted Bases and Fast Transforms Michael Orrison Harvey Mudd College Joint Work with Michael Hansen and Masanori Koyama
Functions Let X = { x 1 , . . . , x n } be a finite set, and let C X be the complex vector space of complex-valued functions defined on X : C X = { f : X → C } . We identify x ∈ X with the function that is 1 on x and 0 on all of other elements. Using the x i as a basis, we can then encode f ∈ C X as a column vector: f ( x 1 ) f ( x 2 ) f �→ . . . . f ( x n ) These column vectors represent the data we would like to analyze.
Actions If G is a finite group acting on X , then G acts on C X where if g ∈ G and f ∈ C X , then ( g · f )( x ) = f ( g − 1 x ) . We can then extend this action to the group algebra C G , which makes C X a C G -permutation module. In the background is a permutation representation ϕ : G → GL | X | ( C ) where ϕ ( g ) is a permutation matrix for all g ∈ G that encodes the action of G on X : � 1 if g · x j = x i [ ϕ ( g )] ij = 0 otherwise .
The Big Idea Let G be a finite group acting on X , and let C X be the resulting permutation module. If we write C X as a direct sum C X = U 1 ⊕ · · · ⊕ U m of submodules, then every f ∈ C X can be written uniquely as f = f 1 + · · · + f m where f i ∈ U i . If the U i are meaningful, then we might be able to better understand f by focusing our attention on the f i . This leads to generalized spectral analysis, which was pioneered by Diaconis.
Example If G = S 3 and X is the set { 1 , 2 , 3 } , then under the usual action of S 3 , we have 1 1 1 , C X = � 1 � ⊕ � − 1 0 � . 1 0 − 1 For example, 11 11 0 20 11 9 = + and = + . 10 11 − 1 3 11 − 8 12 11 1 10 11 − 1
Example If G = X = Z / 4 Z = { 1 , z , z 2 , z 3 } acts on itself by left multiplication, then we have 1 1 1 1 1 i − 1 − i C X = � � ⊕ � � ⊕ � � ⊕ � � . 1 − 1 1 − 1 1 − i − 1 i Note that the associated permutation representation is such that 0 0 0 1 1 0 0 0 z �→ . 0 1 0 0 0 0 1 0
Example If G = S n and X is the set of k -element subsets of { 1 , . . . , n } where k ≤ n / 2, then we can write C X = U 0 ⊕ U 1 ⊕ · · · ⊕ U k where U i corresponds to pure i -th order effects, and f ∈ C X is typically viewed as voting data. Given f = f 0 + f 1 + · · · + f k , we might ask about the extent to which || f i || depends on f 0 , · · · , f i − 1 . (See Algebraic algorithms for sampling from conditional distributions by Diaconis and Sturmfels in Ann. Statist. Volume 26, Number 1 (1998), 363-397.)
Example If G = S 3 and X is the set { 1 , 2 , 3 } , then under the usual action of S 1 , S 2 , and S 3 , we have 1 0 0 C X = � 0 � ⊕ � 1 � ⊕ � 0 � 0 0 1 1 1 0 = � 1 � ⊕ � − 1 � ⊕ � 0 � 0 0 1 1 1 1 , = � 1 � ⊕ � − 1 1 � 1 0 − 2 where these decompositions also reflect the different orbits and a certain kind of adaptedness.
Questions Suppose C X = U 1 ⊕ · · · ⊕ U m where the U i are C G -submodules. 1 Given f ∈ C X , how efficiently can we compute f 1 , . . . , f m ? How efficiently can we compute � f 1 � , . . . , � f m � ? 2 Given a basis B i for each U i , how efficiently can we do a change-of-basis from X to the basis B = B 1 ∪ · · · ∪ B m ? 3 How should the U i and the B i be chosen above so as to be meaningful and also computationally helpful?
Machinery
Discrete Fourier Transforms Every complex group algebra C G is isomorphic to a direct sum of matrix algebras: C G ∼ = C d 1 × d 1 ⊕ · · · ⊕ C d h × d h . Any associated isomorphism D = D 1 ⊕ · · · ⊕ D h is a (generalized) discrete Fourier transform or DFT. Note that the D i form a complete set of irreducible representations for G , and any complete set of irreducible representations for G can be used in this way to construct a DFT for G .
Example � · � · D : C S 3 → C 1 × 1 ⊕ C 1 × 1 ⊕ C 2 × 2 = � � � � · ⊕ · ⊕ · · � 1 � 0 � � � � 1 �→ 1 ⊕ 1 ⊕ 0 1 � 0 � 0 b 1 � � � � 11 �→ 1 ⊕ 0 ⊕ 0 0 � 0 0 � b 3 � � � � 21 �→ 0 ⊕ 0 ⊕ 1 0 � 1 � 0 b 3 11 + b 3 � � � � 22 �→ 0 ⊕ 0 ⊕ 0 1
Subgroup-Adapted DFTs Suppose H ≤ G . The DFT D for G is subgroup-adapted to the chain H ≤ G if for each irreducible representation D i of G and for all h ∈ H , 1 D i ( h ) is block diagonal, where the blocks correspond to irreducible representations of H , and 2 equivalent blocks among all of the D i are actually equal. This can also be extended to longer chains subgroups of G , which we’ll usually take to have the form { 1 } = G 0 < G 1 < · · · < G n = G .
Example = C 1 × 1 ⊕ C 1 × 1 C S 2 ∼ = C 1 × 1 ⊕ C 1 × 1 ⊕ C 2 × 2 C S 3 ∼ D : C S 3 → C 1 × 1 ⊕ C 1 × 1 ⊕ C 2 × 2 � ⋆ � 0 � � � � D | C S 2 : C S 2 → • ⊕ ⋆ ⊕ 0 •
Adapted Bases A basis B for C X is adapted to the DFT D if it can be partitioned B = B 1 ∪ · · · ∪ B k so that each B j spans an irreducible submodule of C X , and if this submodule corresponds to D i , then [ g ] B j = D i ( g ) for all g ∈ G .
Example If G = S 3 and X is the set { 1 , 2 , 3 } , then under the usual action of S 1 , S 2 , and S 3 , we have 1 0 0 C X = � 0 � ⊕ � 1 � ⊕ � 0 � 0 0 1 1 1 0 = � 1 � ⊕ � − 1 � ⊕ � 0 � 0 0 1 1 1 1 , = � 1 � ⊕ � − 1 1 � . 1 0 − 2
Creating Adapted Bases Let D : C G → C d 1 × d 1 ⊕ · · · ⊕ C d h × d h be a DFT. Let b k ij be the unique element in C G such that D ( b k ij ) has zeros everywhere except for a 1 in the ( i , j ) entry of D k . Then the collection { b k ij } of all such elements forms an adapted basis for C G called the dual matrix coefficient basis. More generally, if G acts transitively on X , and x ∈ X , then { g · x } g ∈ G is clearly a spanning set for C X , but { b k ij · x } is spanning set for C X that contains an adapted basis as a subset. This is how we will create adapted bases, but note that the choice of x matters.
Frequency Subspaces The elements of the form b k ii are primitive idempotents for C G , and the subspace b k ii · C X is the associated frequency space of C X . Note that ( b k 11 + · · · + b k d k d k ) · C X is the isotypic subspace corresponding to the representation D k . Key Idea: If D is adapted to the chain H ≤ G , then the frequency spaces of C X with respect to C H are direct sums of the frequency spaces with respect to C G .
Example If G = X = Z / 4 Z = { 1 , z , z 2 , z 3 } and H = 2 Z / 4 Z , then 1 0 1 0 0 1 0 1 C X = � � ⊕ � � ⊕ � � ⊕ � � 1 0 − 1 0 0 1 0 − 1 1 1 1 1 1 − 1 i − i = � � ⊕ � � ⊕ � � ⊕ � � . 1 1 − 1 − 1 1 − 1 − i i
Fast Transforms
Change-of-Basis Suppose D is adapted to the chain H ≤ G . Although G acts on X transitively, the restriction to H need not be transitive, so X might be a union X = X 1 ∪ · · · ∪ X t of orbits of X with respect to H . Suppose you have a basis B ′ for C X that respects this partitition and is adapted to the associated irreducible representations of H . Questions: How difficult is it to do a change-of-basis from B ′ to an adapted basis B with respect to the action of G ? Can we bound the number of nonzero entries in the associated change-of-basis matrix?
Bounds Based on Frequency Subspaces If the frequency spaces with respect to H have dimensions α 1 , . . . , α f , then we will have no more than α 2 1 + · · · + α 2 f nonzero entries in the associated change-of-basis matrix. This can be used to show that if X = G and the X i are the right cosets of H , then the above sum becomes h ([ G : H ] d i ) 2 d i � i =1 where the sum is over all of the irreducible degrees d 1 , . . . , d h .
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