Notation for Basis Vectors
In order to compute the isomorphisms aTTT and aT,T for Fibonacci anyons, we shall view them as matrices, using suitable bases for the relevant vector spaces, and we shall calculate the constraints imposed on those matrices by the pentagon and hexagon conditions. We begin by setting up a convenient notation for those bases.
The domains and codomains of the morphisms under consideration are obtained from т and 1 by iterated ®. We must, of course, be careful about the parenthesization of such ^-products because, as we saw above, different parenthesizations can lead to different bases; indeed, aT,T,T contains exactly the information about how two such bases are related.
In general, given a parenthesized ^-product of т’s and 1’s, we can use the defining equations for Fibonacci anyons, particularly т ® т = 1 ® т, and the distributivity of ® over ®, to convert the given product into a sum of т’s and 1’s. Each summand in that sum arises from the original product as a result of certain choices of 1 or т when expanding some occurrences of т ® т.
For example, in the equation
considered above, the summand 1 at the right end of the equation arose from the т ® (т ® т) at the left end by first choosing the summand т in the evaluation of (т ® т) at the first step in the equation, and then, after applying the distributive law at the second step, choosing the summand 1 in the evaluation of т ® т at the third step. These choices can be visualized as the tree
or, in a more compressed notation,
Here the three т’s and the parentheses describe the ^-product т ® (т ® т) that we began with, and the symbols under the dots indicate the choice of summand at each step. The inner ? indicates that, from the evaluation of the inner т ® т = 1 ® т, we
chose the т summand. After applying distributivity, that leads us to т ® т, from
which, as indicated by the outer ?, we chose the summand 1.
The other possible choices during the same evaluation would be written and depicted by the trees
The first of these indicates that, as before, we chose the т summand when evaluating the inner ®, obtaining, when distributivity is applied, the summand т ® т = 1 ® т, but then we chose the т rather than the 1. The second indicates that, when evaluating the inner т ® т, we chose the summand 1, so that, after applying distributivity, we got т ® 1. Here, there is no choice remaining to be made; т ® 1issimply т. Nevertheless, we write т under the outer dot and at the root of the tree, to make it obvious that the final result here is т.
In what follows, we shall systematically use the compressed notation, but the reader can easily draw the tree diagrams. Indeed, these diagrams are just the parse trees of the compressed notations. The trees can also be viewed as a sort of Feynman diagrams, depicting how the anyons at the leaves of the tree fuse on their way to the root.
In our notation, we write a product of т’s or 1’s, with т ’s or 1’s also under the dots, to represent specific summands (1 or т) in the fully distributed expansion of a ^-product of т’s and 1’s. To evaluate (X • Y), first evaluate X and Y; then apply ®
to them; and then take the 1 summand in the result. To evaluate (X • Y) do the same
except that you take the т summand in the result. These notations will never be used in situations where they would be meaningless because the required summand is not present in the result; that is, we never write (X j Y) when one of X, Y evaluates to 1
and the other to т, for then ® yields only т; and we never write (X • Y) when both of
X, Y evaluate to 1. As in one of the examples above, we include the subscript under the dot even when that subscript is forced because one of the factors evaluates to 1.
Notice that our notation provides symbols, like the three examples above, that denote not only an object 1 or т (which can be read off by just looking under the outermost dot in the notation) but also a particular occurrence of that 1 = (C, 0) or т = (0, C) as a subspace (direct summand) of a specific ^-product, namely the product with the same factors and the same parentheses as in our notation.
In other words, if we are given a parenthesized ^-product of 1’s and т’s, representing the pair of vector spaces (V, Vx), then by replacing each ® by either • or
•, we obtain (either a meaningless expression because some required summand is
absent or) a notation for a subspace of V1 or Vr. It denotes a subspace of V1 (resp. Vx) just in case the outermost ® was replaced by • (resp. •).
Our notation provides names for certain summands 1 = (C, 0) or т = (0, C) of certain objects (Vi, Vx) of the Fibonacci category A. We shall also use the same notation for the resulting basis vectors. That is, once we have a copy of, say, (C, 0) in (V1, Vr), the number 1 in C corresponds to some vector in V1, and we shall use the same notation for this vector as for the summand. The same goes for the case of copies of (0, C) in (V1, Vr); they provide vectors in Vx.
Notice that, if we begin with some parenthesized ^-product of 1’s and т’s, with value (V1, Vт) in A, and if we form all possible (meaningful) notations by replacing ® by • or •, then the resulting vectors, as described in the preceding paragraph,
constitute bases for the vector spaces V1 and Vx. This observation is just a restatement of the fact that the original parenthesized ^-product is the direct sum of all the simple objects obtainable by making the choices indicated by the subscripts in our notation.