# kgp.at Function References¶

## sort_by¶

sort_by:(KGBElt -> int) f->([KGBElt] v) [KGBElt] Defined in line number 51.

Given a list of KGB elements and a function f assigning integers to them, sort the list by weakly increasing value of f

## KGP_elt¶

KGP_elt:KGPElt pair->KGPElt Defined in line number 65.

## S¶

S:KGPElt(S,)->[int] Defined in line number 68.

The list S of simple roots of a KGP element

## root_datum¶

root_datum:KGPElt(,x)->RootDatum Defined in line number 71.

The root datum of the RealForm G of a KGP element

## real_form¶

real_form:KGPElt(,x)->RealForm Defined in line number 74.

The RealForm G of a KGP element

## complement¶

complement:int n,[int] S->[int] Defined in line number 77.

Complement of subset of simple roots in rank n

## find_ascent¶

find_ascent:[int] S, KGBElt x->[KGBElt] Defined in line number 81.

An ascent of x by a generator in S, if any exist

## down_neighbors¶

down_neighbors:[int] S,KGBElt x->[int] Defined in line number 89.

All descents of x by generators in S, there may be duplicates

## is_maximal_in_partial_order¶

is_maximal_in_partial_order:[int] S,KGBElt x->bool Defined in line number 100.

Decide whether x is maximal in the partial order defined by S

## maxima_in_partial_order¶

maxima_in_partial_order:RealForm G,[int] S->[KGBElt] Defined in line number 103.

List maximal KGB elements in the partial order defined by S

## maximal¶

maximal:[int] S, KGBElt x->KGBElt Defined in line number 109.

(unique) maximal element in equivalence class of x

## canonical_representative¶

canonical_representative:KGPElt y->KGPElt Defined in line number 114.

The representative of a KGP element with maximal x

## =¶

=:KGPElt (S,x),KGPElt (T,y)->bool Defined in line number 121.

Equality of KGP elements: (S,x)=(T,y) if these give the same K-orbit of parabolics

## equivalence_class_of¶

equivalence_class_of:KGPElt(S,x):y->[KGBElt] Defined in line number 126.

The equivalence class of a KGB element in partial order defined by S

## x_min¶

x_min:KGPElt P->KGBElt Defined in line number 141.

A minimal KGB element from an equivalence class defined by S (unlike x_max, it is not unique)

## KGP¶

KGP:RealForm G,[int] S->[KGPElt] Defined in line number 146.

The set of KGP elements associated to a RealForm and a set of simple roots S; KGP(G,S) is in bijection with $$K\backslash G/P_S$$

## KGP_numbers¶

KGP_numbers:RealForm G,[int] S->[int] Defined in line number 150.

Just the index numbers (maximal x) of KGP(G,S)

## is_open¶

is_open:KGPElt y->bool Defined in line number 155.

Test whether y in $$K\backslash G/P_S$$ is open: <=> last element of y is last element of KGB

## is_closed¶

is_closed:KGPElt P->bool Defined in line number 158.

Test whether y in $$K\backslash G/P_S$$ is closed: <=> length(first element)=0

## KGP_elt¶

KGP_elt:ratvec lambda,KGBElt x->KGPElt Defined in line number 161.

Parabolic determined by (the stabilizer in W of) a weight lambda

## KGPElt¶

([int], KGBElt) Defined in line number 46.

## Parabolic¶

([int], KGBElt) Defined in line number 47.