springer_table_BC.at

Springer correspondence from nilpotent orbits to representations of W
in type B/C
orbit: partition encoded as semisimple H -> W(C_n)-irrep as pair of partitions
see Carter pg. 419

The function |C_adjust| takes an arbitrary partition of an even number, and
returns a modified version in which the multiplicity over every odd part size
is even. This is achieved by repeatedly moving a square of the diagram from a
row of odd length down to the end of the next row of strictly shorter odd
length (thus creating two new rows of even length, possibly the same length).
Progress is from longer to shorter rows, although given that the number of
rows of odd length must always be even, it could have been defined in to
opposite order with the same result: for the smallest odd row length with odd
multiplicity one square is borrowed from the next larger odd length row, then
repeat for the now smallest odd row length with odd multiplicity,and so on.
All in all a diagram valid for type C is found with minimal distance of
squares moved, though not necessarily with minimal number of squares moved (to
achieve that, odd lengths with even frequency would have to remain untouched).

With respect to |C_adjust|, the function |B_adjust| only differs in that it
assumes a partition of an odd number is given (though this is less essential
for its functioning), and that the multiplicities of nonzero even row lengths
rather then of odd ones are made even, systematically replacing "odd length"
by "even length". The reasoning is somewhat modified, since the number of rows
with nonzero even length can have either parity; however if we need to move
down a box from the last nonzero even length with odd multiplicity, we can
move it to an empty row, thus adding a row of length~$1$. Therefore ordering
the traversal by decreasing row length is essential here. The number of rows
of nonzero length is always made odd, since the number of rows with odd length
must be odd, and that of nonzero even length is made even. To implement, we
manually set the frequency of parts of size zero to $1$, so that the search
for a smaller even length with odd multiplicity cannot fail (if it turns out
to be used, the loop will stop immediately thereafter; if not the frequency of
the zero length remains $1$, but this is ignored by the final |repeat_parts|).

To map a partition of 2n+1 valid for type $B_n$ to a partition of 2n valid for
type $C_n$ we can conceptually first transpose (so every even nonzero column
length has even multiplicity), then remove a square from the last longest row,
run the transposed form of |C_adjust| (which ensures that all odd column
lengths occur with even multiplicity), and finally run the normal |C_adjust|.
This ensures that each maximal group of equal odd length rows starts at an
even index (so has an even number of preceding rows), and ends at an odd one.
To see this, the transposed |C_adjust| has ensured that each odd-indexed row
is shorter that the previous one by an even (possibly zero) amount, from which
one easily deduces that each maximal group of successive odd length rows
starts at an even index and has an even number of rows. The application of
normal |C_adjust| then operates within such groups, and any squares moved are
from an even-indexed row to the next row. As a consequence, the row length
property established by executing the transposed |C_adjust| still holds after
the normal |C_adjust| has run. The diagram of the resulting partition can then
be characterised by the fact that it meets all 2x2 squares of a doubly-sized
grid with the origin as vertex in an even number (0,2, or 4) of squares.

To map to a partition of 2n valid for type $C_n$ to a partition of 2n+1 valid
for type $B_n$, we conceptually first transpose, add a square to the first
column, run the transposed form of |B_adjust| (which ensures that all nonzero
ven column lengths occur with even multiplicity), and finally run the normal
|B_adjust|. This time row 0 has odd length, and following pairs of odd-index
and successive even-index rows have lengths of equal parity. Again every
square that moves is between two such successive rows, and the property
established by the transposed |B_adjust| is preserved by the normal
|B_adjust|. The diagram of the resulting partition can then be characterised
by the fact that it meets all 2x2 squares of a doubly-sized grid with $(1,1)$
as vertex, with the exception of the 2x2 square centered at the origin, in an
even number of squares

the dual maps are defined in terms of the |diagram| or |stratified_diagram| of
|O|, thus ignoring, and they should any radical component of |O.H|