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D. Gómez-Ullate et al.
4 Categorical Structure
In this section, we define MD, a category whose objects are Maya diagrams and
whose arrows are multi-flips, and REXT, another category whose objects are
rational extensions and whose arrows are intertwining operators (definition given
below). We then exhibit a functor MD → REXT that we use to classify ladder
operators.
In order to define composition of arrows, it will first be necessary to generalize
the notion of a multi-flip. A multi-set is a generalized set object that allows for
multiple instances of each of its elements. Let
Z p denote the set of integer multisets of cardinality p and
Z =
p
Z p the set of finite integer multi-sets. We express
a multi-set K ∈
Z as
K = {k
p 1
1 , . . . , k
p q
q }
(12)
where k 1 , . . . , k q ∈ Z are distinct, and where p i > 0 indicate the multiplicity of
element k i . The cardinality is then given by p = p 1 + · · · + p q . The notion of a
multi-flip extends naturally from sets to multi-sets. Indeed, for K ∈
Z, we re-use (2)
to define the multi-flip f K : M → M.
We say that K is an even multi-set if all of its elements have an even multiplicity.
Since flips are involutions, f K is the identity transformation if and only if K is
even. If K is an even multi-set, then it has the unique decomposition K = K 1 ∪ K 1
where K 1 has the same elements as K but with the multiplicities divided by 2. More
generally, every multi-set K ∈
Z has a unique decomposition of the form
K = K 0 ∪ K 1 ∪ K 1 , K 0 ∈ Z, K 1 ∈
Z,
(13)
where K 0 is the set of integers that occur in K with an odd multiplicity. Again, since
flips are involutions, we have f K = f K 0 .
The objects of MD are labelled Maya diagrams M, and the arrows are pairs
(M, K) ∈ M ×
Z. The source of (M, K) is M, and the target is f K (M).
Composition of morphisms is given by the union of multi-sets:
(M 2 , K 2 ) ◦ (M 1 , K 1 ) = (M 1 , K 1 ∪ K 2 ),
where M 1 ∈ M, K 1 , K 2 ∈
Z, M 2 = f K 1 (M 1 ).
For differential operators A, T 1 , T 2 , we say that A intertwines T 1 , T 2 if
AT 1 = T 2 A.
The objects of REXT are the rational extensions T M , M ∈ M, and the arrows
are monic differential operators that intertwine two rational extensions. Observe
that if A intertwines T 1 , T 2 , then so does A ◦ p(T 1 ), where p(x) is an arbitrary
D. Gómez-Ullate et al.
4 Categorical Structure
In this section, we define MD, a category whose objects are Maya diagrams and
whose arrows are multi-flips, and REXT, another category whose objects are
rational extensions and whose arrows are intertwining operators (definition given
below). We then exhibit a functor MD → REXT that we use to classify ladder
operators.
In order to define composition of arrows, it will first be necessary to generalize
the notion of a multi-flip. A multi-set is a generalized set object that allows for
multiple instances of each of its elements. Let
Z p denote the set of integer multisets of cardinality p and
Z =
p
Z p the set of finite integer multi-sets. We express
a multi-set K ∈
Z as
K = {k
p 1
1 , . . . , k
p q
q }
(12)
where k 1 , . . . , k q ∈ Z are distinct, and where p i > 0 indicate the multiplicity of
element k i . The cardinality is then given by p = p 1 + · · · + p q . The notion of a
multi-flip extends naturally from sets to multi-sets. Indeed, for K ∈
Z, we re-use (2)
to define the multi-flip f K : M → M.
We say that K is an even multi-set if all of its elements have an even multiplicity.
Since flips are involutions, f K is the identity transformation if and only if K is
even. If K is an even multi-set, then it has the unique decomposition K = K 1 ∪ K 1
where K 1 has the same elements as K but with the multiplicities divided by 2. More
generally, every multi-set K ∈
Z has a unique decomposition of the form
K = K 0 ∪ K 1 ∪ K 1 , K 0 ∈ Z, K 1 ∈
Z,
(13)
where K 0 is the set of integers that occur in K with an odd multiplicity. Again, since
flips are involutions, we have f K = f K 0 .
The objects of MD are labelled Maya diagrams M, and the arrows are pairs
(M, K) ∈ M ×
Z. The source of (M, K) is M, and the target is f K (M).
Composition of morphisms is given by the union of multi-sets:
(M 2 , K 2 ) ◦ (M 1 , K 1 ) = (M 1 , K 1 ∪ K 2 ),
where M 1 ∈ M, K 1 , K 2 ∈
Z, M 2 = f K 1 (M 1 ).
For differential operators A, T 1 , T 2 , we say that A intertwines T 1 , T 2 if
AT 1 = T 2 A.
The objects of REXT are the rational extensions T M , M ∈ M, and the arrows
are monic differential operators that intertwine two rational extensions. Observe
that if A intertwines T 1 , T 2 , then so does A ◦ p(T 1 ), where p(x) is an arbitrary