Ladder Operators and Rational Extensions
123
It is useful to regard M as a complete graph whose edges are multi-flips. For
Maya diagrams M 1 , M 2 ∈ M, the symmetric difference
M 1 M 2 = (M 1 \ M 2 ) ∪ (M 2 \ M 1 )
is precisely the edge that connects M 1 and M 2 . More precisely, if
K = M 1 M 2 = M 2 M 1 ,
then f K (M 1 ) = M 2 and f K (M 2 ) = M 1 .
Multi-flips can also be used to define a bijection Z → M given by K →
f K (M ∅ ), where M ∅ := Z − denotes the trivial Maya diagram. We refer to K ∈ Z as
the index set of the Maya diagram f K (M ∅ ).
The additive group Z acts on M, because for M ∈ M and n ∈ Z, the set
M + n = {m + n : m ∈ M}
is also a Maya diagram. Moreover, we have
σ M+n = σ M + n.
(3)
We will refer to an equivalence class of Maya diagrams related by translations as
an unlabelled Maya diagram, and denote the set of all unlabelled Maya diagrams by
M/Z. One can visualize the passage from an unlabelled to a labelled Maya diagram
as choosing the placement of the origin.
For B ∈ Z p , where p = 2g + 1 is odd, we define the Maya diagram
Ξ(B) = (−∞, b 0 ) ∪ [b 1 , b 2 ) ∪ · · · ∪ [b 2g−1 , b 2g ),
(4)
where b 0 < b 1 < · · · < b 2g is an increasing enumeration of B and where [m, n) =
{j ∈ Z : m ≤ j < n}. Every Maya diagram has a unique representation of the form
Ξ(B) for some B ∈ Z 2g+1 . We will call the corresponding g ≥ 0 the genus of
M = Ξ(B) and refer to (b 0 , . . . , b 2g ) as the block coordinates of M. The block
coordinates may also be characterized as the unique set B ∈ Z such that f B (M) =
M + 1.
After removal of the initial infinite •
segment and the trailing infinite segment,
a Maya diagram consists of alternating empty and filled •
segments of variable
length. The genus g counts the number of such pairs. The even block coordinates b 2i
indicate the starting positions of the empty segments, and the odd block coordinates
b 2i+1 indicate the starting positions of the filled segments.
123
It is useful to regard M as a complete graph whose edges are multi-flips. For
Maya diagrams M 1 , M 2 ∈ M, the symmetric difference
M 1 M 2 = (M 1 \ M 2 ) ∪ (M 2 \ M 1 )
is precisely the edge that connects M 1 and M 2 . More precisely, if
K = M 1 M 2 = M 2 M 1 ,
then f K (M 1 ) = M 2 and f K (M 2 ) = M 1 .
Multi-flips can also be used to define a bijection Z → M given by K →
f K (M ∅ ), where M ∅ := Z − denotes the trivial Maya diagram. We refer to K ∈ Z as
the index set of the Maya diagram f K (M ∅ ).
The additive group Z acts on M, because for M ∈ M and n ∈ Z, the set
M + n = {m + n : m ∈ M}
is also a Maya diagram. Moreover, we have
σ M+n = σ M + n.
(3)
We will refer to an equivalence class of Maya diagrams related by translations as
an unlabelled Maya diagram, and denote the set of all unlabelled Maya diagrams by
M/Z. One can visualize the passage from an unlabelled to a labelled Maya diagram
as choosing the placement of the origin.
For B ∈ Z p , where p = 2g + 1 is odd, we define the Maya diagram
Ξ(B) = (−∞, b 0 ) ∪ [b 1 , b 2 ) ∪ · · · ∪ [b 2g−1 , b 2g ),
(4)
where b 0 < b 1 < · · · < b 2g is an increasing enumeration of B and where [m, n) =
{j ∈ Z : m ≤ j < n}. Every Maya diagram has a unique representation of the form
Ξ(B) for some B ∈ Z 2g+1 . We will call the corresponding g ≥ 0 the genus of
M = Ξ(B) and refer to (b 0 , . . . , b 2g ) as the block coordinates of M. The block
coordinates may also be characterized as the unique set B ∈ Z such that f B (M) =
M + 1.
After removal of the initial infinite •
segment and the trailing infinite segment,
a Maya diagram consists of alternating empty and filled •
segments of variable
length. The genus g counts the number of such pairs. The even block coordinates b 2i
indicate the starting positions of the empty segments, and the odd block coordinates
b 2i+1 indicate the starting positions of the filled segments.