Ladder Operators and Rational Extensions
127
polynomial. Given T 1 , T 2 , we say that A is a primitive intertwiner if it does not
include a nontrivial right factor p(T 1 ).
For a Maya diagram M ∈ M and a set K ∈ Z p , we define the operator
A M,K [y] =
Wr[ψ M,k 1 , . . . , ψ M,k p , y]
Wr[ψ M,k 1 , . . . , ψ M,k p ]
.
By construction, A M,K is a monic differential operator of order p. These intertwining operators have their origin in SUSYQM (supersymmetric quantum mechanics),
and obey the intertwining relation
A M 1 ,K T M 1 = T M 2 A M 1 ,K , M 2 = f K (M 1 ), M 1 , M 2 ∈ M, K ∈ Z.
It is possible to show that A M,K is a primitive intertwiner between T M and
T f K (M) . Moreover, it is possible to show [2, Proof of Theorem 3.10] that every
arrow in REXT has the form A M,K ◦ p(T M ), where A M,K is primitive (i.e., K is a
set), and p(x) is a polynomial. We also note that these intertwiners are translation
invariant:
A M+n,K+n = A M,K , n ∈ Z.
(14)
In order to describe the composition of intertwiners, we need to extend the above
definition to include multi-sets. For K ∈
Z, let K 0 ∈ Z and K 1 ∈
Z be as per (13).
For M ∈ M, we now define
A M,K = A M,K 0 ◦
k∈K 1
(2k + 1 − T M ).
(15)
In other words, if K ∈
Z contains elements of higher multiplicity, then A M,K is no
longer primitive. The arrows of REXT are the operators A M,K , M ∈ M, K ∈
Z.
Composition of arrows is just the usual composition of differential operators.
Theorem 1 The correspondence M → T M , M ∈ M and (M, K) → A M,K , K ∈
Z is a covariant functor MD → REXT.
Proof It suffices to observe that for M 1 ∈ M, K 1 , K 2 ∈
Z we have
A M 2 ,K 2 ◦ A M 1 ,K 1 = A M 1 ,K 1 ∪K 2 , M 2 = f K 1 (M 1 ).
5 Ladder Operators
We define a ladder operator to be an intertwiner A such that
AT M = (T M + λ)A
127
polynomial. Given T 1 , T 2 , we say that A is a primitive intertwiner if it does not
include a nontrivial right factor p(T 1 ).
For a Maya diagram M ∈ M and a set K ∈ Z p , we define the operator
A M,K [y] =
Wr[ψ M,k 1 , . . . , ψ M,k p , y]
Wr[ψ M,k 1 , . . . , ψ M,k p ]
.
By construction, A M,K is a monic differential operator of order p. These intertwining operators have their origin in SUSYQM (supersymmetric quantum mechanics),
and obey the intertwining relation
A M 1 ,K T M 1 = T M 2 A M 1 ,K , M 2 = f K (M 1 ), M 1 , M 2 ∈ M, K ∈ Z.
It is possible to show that A M,K is a primitive intertwiner between T M and
T f K (M) . Moreover, it is possible to show [2, Proof of Theorem 3.10] that every
arrow in REXT has the form A M,K ◦ p(T M ), where A M,K is primitive (i.e., K is a
set), and p(x) is a polynomial. We also note that these intertwiners are translation
invariant:
A M+n,K+n = A M,K , n ∈ Z.
(14)
In order to describe the composition of intertwiners, we need to extend the above
definition to include multi-sets. For K ∈
Z, let K 0 ∈ Z and K 1 ∈
Z be as per (13).
For M ∈ M, we now define
A M,K = A M,K 0 ◦
k∈K 1
(2k + 1 − T M ).
(15)
In other words, if K ∈
Z contains elements of higher multiplicity, then A M,K is no
longer primitive. The arrows of REXT are the operators A M,K , M ∈ M, K ∈
Z.
Composition of arrows is just the usual composition of differential operators.
Theorem 1 The correspondence M → T M , M ∈ M and (M, K) → A M,K , K ∈
Z is a covariant functor MD → REXT.
Proof It suffices to observe that for M 1 ∈ M, K 1 , K 2 ∈
Z we have
A M 2 ,K 2 ◦ A M 1 ,K 1 = A M 1 ,K 1 ∪K 2 , M 2 = f K 1 (M 1 ).
5 Ladder Operators
We define a ladder operator to be an intertwiner A such that
AT M = (T M + λ)A