The Efficient Computation of Fourier Transforms on Semisimple Algebras

Abstract

We present a general diagrammatic approach to the construction of efficient algorithms for computing a Fourier transform on a semisimple algebra. This extends previous work wherein we derive best estimates for the computation of a Fourier transform for a large class of finite groups. We continue to find efficiencies by exploiting a connection between Bratteli diagrams and the derived path algebra and construction of Gel’fand–Tsetlin bases. Particular results include highly efficient algorithms for the Brauer, Temperley–Lieb, and Birman–Murakami–Wenzl algebras.

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Notes

  1. 1.

    We use here a standard definition of operation count as a complex addition and multiplication. In various places we may break out the number of additions and multiplications separately, but this will have no effect on the “big O” kinds of results we present here.

  2. 2.

    A straight-line program is a list of instructions for performing the operations \(\times , \div , +, -\) on inputs and precomputed values [7].

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Acknowledgements

Daniel N. Rockmore was partially supported by AFOSR Award FA9550-11-1-0166 and the Neukom Institute for Computational Science at Dartmouth College. Sarah Wolff was partially supported by an NSF Graduate Fellowship.

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Correspondence to Sarah Wolff.

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Communicated by Thomas Strohmer.

Appendices

Appendix A: Brauer Algebra Combinatorial Lemmas

Let \(\mathcal {B}\) denote the Bratteli diagram associated to the chain of Brauer algebras \(\mathcal {B}r_n>\mathcal {B}r_{n-1}>\cdots \mathcal {B}r_1>\mathcal {B}r_0\) (Fig. 1). The following two lemmas provide a bound for \(\#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})\), for \(\mathcal {H}_i^n\) as in Fig. 12.

Lemma A.1

  1. (1)

    \(\displaystyle \#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})=\frac{ \dim (\mathcal {B}r_{n-1})}{\dim (\mathcal {B}r_{i-1})}\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B}),\)

  2. (2)

    \(\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})\)

    $$\begin{aligned} \le 2\displaystyle \frac{\dim (\mathcal {B}r_{i-1})^2}{\dim (\mathcal {B}r_{i-2})}+\sum _{\beta _{i-1}\in \mathcal {B}^{i-1}}(4{\text {jmp}}(\beta _{i-1})^2+2{\text {jmp}}(\beta _{i-1})+1)(d_{\beta _{i-1}})^2, \end{aligned}$$

where \({\text {jmp}}\) denotes the jump of a partition, i.e, the number of ways to remove a single box to form a new partition.

Proof

Part (1) has the same proof as Lemma D.3 in [35].

To prove (2), consider

$$\begin{aligned} \begin{array}{l}\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})\\ =\displaystyle \sum _{\alpha _j,\beta _j\in \mathcal {B}^j}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i},\alpha _{i-1})M_\mathcal {B}(\alpha _{i},\beta _{i-1})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-1}}d_{\alpha _{i-1}}\\ =\displaystyle \sum _{\alpha _{i-1}\ne \beta _{i-1}}+\sum _{\alpha _{i-1}=\beta _{i-1}},\end{array} \end{aligned}$$

for \(\displaystyle \sum _{\alpha _{i-1}\ne \beta _{i-1}}\) the sum

$$\begin{aligned} \begin{array}{l}\displaystyle \sum _{\begin{array}{c} \alpha _j,\beta _j\in \mathcal {B}^j\\ \alpha _{i-1}\ne \beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})M_\mathcal {B}(\alpha _{i},\alpha _{i-1})M_\mathcal {B}(\alpha _{i},\beta _{i-1})M_\mathcal {B}(\alpha _{i-1},\beta _{i-2})d_{\beta _{i-1}}d_{\alpha _{i-1}}\end{array} \end{aligned}$$

and \(\displaystyle \sum _{\alpha _{i-1}=\beta _{i-1}}\) the sum

$$\begin{aligned} \begin{array}{l}\displaystyle \sum _{\begin{array}{c} \alpha _j,\beta _j\in \mathcal {B}^j\\ \alpha _{i-1}=\beta _{i-1} \end{array}}M_\mathcal {B}(\beta _{i-1},\beta _{i-2})^2M_\mathcal {B}(\alpha _{i},\beta _{i-1})^2(d_{\beta _{i-1}})^2.\end{array} \end{aligned}$$

First suppose \(\alpha _{i-1}\) and \(\beta _{i-1}\) are distinct partitions. Then they jointly determine \(\alpha _i\) up to two choices. This is clear if \(\alpha _{i-1}\) and \(\beta _{i-1}\) both partition k, as they then jointly determine exactly one partition of \(k+1\) and one partition of \(k-1\). Now suppose, without loss of generality, that \(\alpha _{i-1}\) is a partition of k while \(\beta _{i-1}\) is a partition of \(k-2\). Then to both be connected to a vertex, \(\alpha _i\), at level i, \(\beta _{i-1}\) must be obtained from \(\alpha _{i-1}\) by removing two boxes, which can only be done in two ways.

Then as in the proof of Lemma D.3 of [35],

$$\begin{aligned} \sum _{\alpha _{i-1}\ne \beta _{i-1}}\le 2\left( \frac{\dim (\mathcal {B}r_{i-1})^2}{\dim (\mathcal {B}r_{i-2})}-\sum _{}{\text {jmp}}(\beta _{i-1})(d_{\beta _{i-1}})^2\right) . \end{aligned}$$
(5)

Now suppose \(\alpha _{i-1}=\beta _{i-1}\). Then \(\alpha _i\) is obtained from \(\beta _{i-1}\) by either adding or removing a box, and similarly for \(\beta _{i-2}\). Thus,

$$\begin{aligned} \sum _{\alpha _{i-1}=\beta _{i-1}}=\sum _{\beta _{i-1}\in \mathcal {B}^{i-1}}(2{\text {jmp}}(\beta _{i-1})+1)(2{\text {jmp}}(\beta _{i-1})+1)(d_{\beta _{i-1}})^2. \end{aligned}$$
(6)

Summing Eqs. (5) and (6) gives part (2). \(\square \)

Combining Lemma A.1 with the fact that \({\text {jmp}}(\beta _i)^2\le 2i\) (see proof of [30, Lemma 5.3]) gives the following bound:

Corollary A.2

\(\#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})\le \frac{16i-17}{2n-1}\dim (\mathcal {B}r_n)\).

Appendix B: Temperley–Lieb Algebra Combinatorial Lemmas

Let \(\mathcal {B}\) denote the Bratteli diagram associated to the chain of Temperley–Lieb algebras \(\mathcal {T}_n>\mathcal {T}_{n-1}>\cdots \mathcal {T}_1>\mathcal {T}_0\) (Fig. 13). The following two lemmas provide a bound for \(\#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})\), for \(\mathcal {H}_i^n\) as in Fig. 12.

Lemma B.1

  1. (1)

    \(\displaystyle \#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})=\frac{ \dim (\mathcal {T}_{n-1})}{\dim (\mathcal {T}_{i-1})}\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B}),\)

  2. (2)

    \(\#{\text {Hom}}(\mathcal {H}_i^i\uparrow G;\mathcal {B})\le \displaystyle \frac{\dim (\mathcal {T}_{i-1})^2}{\dim (\mathcal {T}_{i-2})}+\sum _{\beta _{i-1}\in \mathcal {B}^{i-1}}({\text {jmp}}(\beta _{i-1})^2(d_{\beta _{i-1}})^2.\)

Proof

This is exactly Lemma 5.2 of [30], replacing the order of the symmetric group with the dimension of the Temperley–Lieb algebra. \(\square \)

Combining Lemma A.1 with the fact that \({\text {jmp}}(\beta _i)^2\le 2i\)

Corollary B.2

\(\#{\text {Hom}}(\mathcal {H}_i^n\uparrow G;\mathcal {B})\le \frac{(4i-6+2i^2)(n+1)(n)}{i(2n)(2n-1)}\dim (\mathcal {T}_n)\).

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Maslen, D., Rockmore, D.N. & Wolff, S. The Efficient Computation of Fourier Transforms on Semisimple Algebras. J Fourier Anal Appl 24, 1377–1400 (2018). https://doi.org/10.1007/s00041-017-9555-5

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Keywords

  • Fast Fourier transform
  • Bratteli diagram
  • Path algebra
  • Quiver

Mathematics Subject Classification

  • 65250
  • 43A30
  • 05E40
  • 20C15