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.

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

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

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

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

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

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

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