{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE ScopedTypeVariables #-}
module Math.Algebra.Jack
(jack', zonal', schur', skewSchur', jack, zonal, schur, skewSchur)
where
import Prelude
hiding ((*), (+), (-), (/), (^), (*>), product, sum, fromIntegral, fromInteger)
import Algebra.Additive ( (+), (-), sum, zero )
import Algebra.Ring ( (*), product, one, (^), fromInteger )
import Algebra.ToInteger ( fromIntegral )
import qualified Algebra.Field as AlgField
import qualified Algebra.Ring as AlgRing
import Control.Lens ( (.~), element )
import Data.Array ( Array, (!), (//), listArray )
import Data.Maybe ( fromJust, isJust )
import qualified Data.Map.Strict as DM
import Math.Algebra.Jack.Internal ( (.^), _N, jackCoeffC
, jackCoeffP, jackCoeffQ
, _betaratio, _isPartition
, Partition, skewSchurLRCoefficients
, isSkewPartition, _fromInt )
jack'
:: [Rational]
-> Partition
-> Rational
-> Char
-> Rational
jack' :: [Rational] -> Partition -> Rational -> Char -> Rational
jack' = [Rational] -> Partition -> Rational -> Char -> Rational
forall a. C a => [a] -> Partition -> a -> Char -> a
jack
jack :: forall a. AlgField.C a
=> [a]
-> Partition
-> a
-> Char
-> a
jack :: forall a. C a => [a] -> Partition -> a -> Char -> a
jack [] Partition
_ a
_ Char
_ = [Char] -> a
forall a. HasCallStack => [Char] -> a
error [Char]
"jack: empty list of variables"
jack x :: [a]
x@(a
x0:[a]
_) Partition
lambda a
alpha Char
which =
case Partition -> Bool
_isPartition Partition
lambda of
Bool
False -> [Char] -> a
forall a. HasCallStack => [Char] -> a
error [Char]
"jack: invalid integer partition"
Bool
True -> case Char
which of
Char
'J' -> a
resultJ
Char
'C' -> Partition -> a -> a
forall a. C a => Partition -> a -> a
jackCoeffC Partition
lambda a
alpha a -> a -> a
forall a. C a => a -> a -> a
* a
resultJ
Char
'P' -> Partition -> a -> a
forall a. C a => Partition -> a -> a
jackCoeffP Partition
lambda a
alpha a -> a -> a
forall a. C a => a -> a -> a
* a
resultJ
Char
'Q' -> Partition -> a -> a
forall a. C a => Partition -> a -> a
jackCoeffQ Partition
lambda a
alpha a -> a -> a
forall a. C a => a -> a -> a
* a
resultJ
Char
_ -> [Char] -> a
forall a. HasCallStack => [Char] -> a
error [Char]
"jack: please use 'J', 'C', 'P' or 'Q' for last argument"
where
resultJ :: a
resultJ = Int
-> Int
-> Partition
-> Partition
-> Array (Int, Int) (Maybe a)
-> a
-> a
jac ([a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
x) Int
0 Partition
lambda Partition
lambda Array (Int, Int) (Maybe a)
forall {a}. Array (Int, Int) (Maybe a)
arr0 a
forall a. C a => a
one
nll :: Int
nll = Partition -> Partition -> Int
_N Partition
lambda Partition
lambda
n :: Int
n = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
x
arr0 :: Array (Int, Int) (Maybe a)
arr0 = ((Int, Int), (Int, Int)) -> [Maybe a] -> Array (Int, Int) (Maybe a)
forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray ((Int
1, Int
1), (Int
nll, Int
n)) (Int -> Maybe a -> [Maybe a]
forall a. Int -> a -> [a]
replicate (Int
nll Int -> Int -> Int
forall a. C a => a -> a -> a
* Int
n) Maybe a
forall a. Maybe a
Nothing)
theproduct :: Int -> a
theproduct :: Int -> a
theproduct Int
nu0 = if Int
nu0 Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
1
then a
forall a. C a => a
one
else [a] -> a
forall a. C a => [a] -> a
product ([a] -> a) -> [a] -> a
forall a b. (a -> b) -> a -> b
$ (Int -> a) -> Partition -> [a]
forall a b. (a -> b) -> [a] -> [b]
map (\Int
i -> a
forall a. C a => a
one a -> a -> a
forall a. C a => a -> a -> a
+ Int
i Int -> a -> a
forall a. C a => Int -> a -> a
.^ a
alpha) [Int
1 .. Int
nu0Int -> Int -> Int
forall a. C a => a -> a -> a
-Int
1]
jac :: Int -> Int -> [Int] -> [Int] -> Array (Int,Int) (Maybe a) -> a -> a
jac :: Int
-> Int
-> Partition
-> Partition
-> Array (Int, Int) (Maybe a)
-> a
-> a
jac Int
m Int
k Partition
mu Partition
nu Array (Int, Int) (Maybe a)
arr a
beta
| Partition -> Bool
forall a. [a] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null Partition
nu Bool -> Bool -> Bool
|| Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!Int
0 Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 Bool -> Bool -> Bool
|| Int
m Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = a
forall a. C a => a
one
| Partition -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length Partition
nu Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
m Bool -> Bool -> Bool
&& Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!Int
m Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0 = a
forall a. C a => a
zero
| Int
m Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1 = a
x0 a -> Integer -> a
forall a. C a => a -> Integer -> a
^ (Int -> Integer
forall a b. (C a, C b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!Int
0) a -> a -> a
forall a. C a => a -> a -> a
* Int -> a
theproduct (Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!Int
0)
| Int
k Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 Bool -> Bool -> Bool
&& Maybe a -> Bool
forall a. Maybe a -> Bool
isJust (Array (Int, Int) (Maybe a)
arr Array (Int, Int) (Maybe a) -> (Int, Int) -> Maybe a
forall i e. Ix i => Array i e -> i -> e
! (Partition -> Partition -> Int
_N Partition
lambda Partition
nu, Int
m)) =
Maybe a -> a
forall a. HasCallStack => Maybe a -> a
fromJust (Maybe a -> a) -> Maybe a -> a
forall a b. (a -> b) -> a -> b
$ Array (Int, Int) (Maybe a)
arr Array (Int, Int) (Maybe a) -> (Int, Int) -> Maybe a
forall i e. Ix i => Array i e -> i -> e
! (Partition -> Partition -> Int
_N Partition
lambda Partition
nu, Int
m)
| Bool
otherwise = a
s
where
s :: a
s = a -> Int -> a
go (Int
-> Int
-> Partition
-> Partition
-> Array (Int, Int) (Maybe a)
-> a
-> a
jac (Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) Int
0 Partition
nu Partition
nu Array (Int, Int) (Maybe a)
arr a
forall a. C a => a
one a -> a -> a
forall a. C a => a -> a -> a
* a
beta a -> a -> a
forall a. C a => a -> a -> a
* [a]
x[a] -> Int -> a
forall a. HasCallStack => [a] -> Int -> a
!!(Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) a -> Integer -> a
forall a. C a => a -> Integer -> a
^ (Int -> Integer
forall a b. (C a, C b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Partition -> Int
forall a. C a => [a] -> a
sum Partition
mu Int -> Int -> Int
forall a. C a => a -> a -> a
- Partition -> Int
forall a. C a => [a] -> a
sum Partition
nu))
(Int -> Int -> Int
forall a. Ord a => a -> a -> a
max Int
1 Int
k)
go :: a -> Int -> a
go :: a -> Int -> a
go !a
ss Int
ii
| Partition -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length Partition
nu Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
ii Bool -> Bool -> Bool
|| Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!(Int
iiInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = a
ss
| Bool
otherwise =
let u :: Int
u = Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!(Int
iiInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) in
if Partition -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length Partition
nu Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
ii Bool -> Bool -> Bool
&& Int
u Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0 Bool -> Bool -> Bool
|| Int
u Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!Int
ii
then
let nu' :: Partition
nu' = (Int -> IndexedTraversal' Int Partition Int
forall (t :: * -> *) a.
Traversable t =>
Int -> IndexedTraversal' Int (t a) a
element (Int
iiInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) ((Int -> Identity Int) -> Partition -> Identity Partition)
-> Int -> Partition -> Partition
forall s t a b. ASetter s t a b -> b -> s -> t
.~ Int
uInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) Partition
nu in
let gamma :: a
gamma = a
beta a -> a -> a
forall a. C a => a -> a -> a
* Partition -> Partition -> Int -> a -> a
forall a. C a => Partition -> Partition -> Int -> a -> a
_betaratio Partition
mu Partition
nu Int
ii a
alpha in
if Int
u Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
1
then
a -> Int -> a
go (a
ss a -> a -> a
forall a. C a => a -> a -> a
+ Int
-> Int
-> Partition
-> Partition
-> Array (Int, Int) (Maybe a)
-> a
-> a
jac Int
m Int
ii Partition
mu Partition
nu' Array (Int, Int) (Maybe a)
arr a
gamma) (Int
ii Int -> Int -> Int
forall a. C a => a -> a -> a
+ Int
1)
else
if Partition
nu' Partition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!! Int
0 Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0
then
a -> Int -> a
go (a
ss a -> a -> a
forall a. C a => a -> a -> a
+ a
gamma a -> a -> a
forall a. C a => a -> a -> a
* [a]
x[a] -> Int -> a
forall a. HasCallStack => [a] -> Int -> a
!!(Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1)a -> Integer -> a
forall a. C a => a -> Integer -> a
^ (Int -> Integer
forall a b. (C a, C b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Partition -> Int
forall a. C a => [a] -> a
sum Partition
mu)) (Int
ii Int -> Int -> Int
forall a. C a => a -> a -> a
+ Int
1)
else
let arr' :: Array (Int, Int) (Maybe a)
arr' = Array (Int, Int) (Maybe a)
arr Array (Int, Int) (Maybe a)
-> [((Int, Int), Maybe a)] -> Array (Int, Int) (Maybe a)
forall i e. Ix i => Array i e -> [(i, e)] -> Array i e
// [((Partition -> Partition -> Int
_N Partition
lambda Partition
nu, Int
m), a -> Maybe a
forall a. a -> Maybe a
Just a
ss)] in
let jck :: a
jck = Int
-> Int
-> Partition
-> Partition
-> Array (Int, Int) (Maybe a)
-> a
-> a
jac (Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) Int
0 Partition
nu' Partition
nu' Array (Int, Int) (Maybe a)
arr' a
forall a. C a => a
one in
let jck' :: a
jck' = a
jck a -> a -> a
forall a. C a => a -> a -> a
* a
gamma a -> a -> a
forall a. C a => a -> a -> a
*
[a]
x[a] -> Int -> a
forall a. HasCallStack => [a] -> Int -> a
!!(Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) a -> Integer -> a
forall a. C a => a -> Integer -> a
^ (Int -> Integer
forall a b. (C a, C b) => a -> b
fromIntegral (Int -> Integer) -> Int -> Integer
forall a b. (a -> b) -> a -> b
$ Partition -> Int
forall a. C a => [a] -> a
sum Partition
mu Int -> Int -> Int
forall a. C a => a -> a -> a
- Partition -> Int
forall a. C a => [a] -> a
sum Partition
nu') in
a -> Int -> a
go (a
ss a -> a -> a
forall a. C a => a -> a -> a
+ a
jck') (Int
ii Int -> Int -> Int
forall a. C a => a -> a -> a
+ Int
1)
else
a -> Int -> a
go a
ss (Int
ii Int -> Int -> Int
forall a. C a => a -> a -> a
+ Int
1)
zonal'
:: [Rational]
-> Partition
-> Rational
zonal' :: [Rational] -> Partition -> Rational
zonal' = [Rational] -> Partition -> Rational
forall a. C a => [a] -> Partition -> a
zonal
zonal :: AlgField.C a
=> [a]
-> Partition
-> a
zonal :: forall a. C a => [a] -> Partition -> a
zonal [a]
x Partition
lambda = Partition -> a -> a
forall a. C a => Partition -> a -> a
jackCoeffC Partition
lambda a
alpha a -> a -> a
forall a. C a => a -> a -> a
* [a] -> Partition -> a -> Char -> a
forall a. C a => [a] -> Partition -> a -> Char -> a
jack [a]
x Partition
lambda a
alpha Char
'J'
where
alpha :: a
alpha = Integer -> a
forall a. C a => Integer -> a
fromInteger Integer
2
schur'
:: [Rational]
-> Partition
-> Rational
schur' :: [Rational] -> Partition -> Rational
schur' = [Rational] -> Partition -> Rational
forall a. C a => [a] -> Partition -> a
schur
schur :: forall a. AlgRing.C a
=> [a]
-> Partition
-> a
schur :: forall a. C a => [a] -> Partition -> a
schur [] Partition
_ = [Char] -> a
forall a. HasCallStack => [Char] -> a
error [Char]
"schur: empty list of variables"
schur x :: [a]
x@(a
x0:[a]
_) Partition
lambda =
case Partition -> Bool
_isPartition Partition
lambda of
Bool
False -> [Char] -> a
forall a. HasCallStack => [Char] -> a
error [Char]
"schur: invalid integer partition"
Bool
True -> Int -> Int -> Partition -> Array (Int, Int) (Maybe a) -> a
sch Int
n Int
1 Partition
lambda Array (Int, Int) (Maybe a)
forall {a}. Array (Int, Int) (Maybe a)
arr0
where
nll :: Int
nll = Partition -> Partition -> Int
_N Partition
lambda Partition
lambda
n :: Int
n = [a] -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length [a]
x
arr0 :: Array (Int, Int) (Maybe a)
arr0 = ((Int, Int), (Int, Int)) -> [Maybe a] -> Array (Int, Int) (Maybe a)
forall i e. Ix i => (i, i) -> [e] -> Array i e
listArray ((Int
1, Int
1), (Int
nll, Int
n)) (Int -> Maybe a -> [Maybe a]
forall a. Int -> a -> [a]
replicate (Int
nll Int -> Int -> Int
forall a. C a => a -> a -> a
* Int
n) Maybe a
forall a. Maybe a
Nothing)
sch :: Int -> Int -> [Int] -> Array (Int,Int) (Maybe a) -> a
sch :: Int -> Int -> Partition -> Array (Int, Int) (Maybe a) -> a
sch Int
m Int
k Partition
nu Array (Int, Int) (Maybe a)
arr
| Partition -> Bool
forall a. [a] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null Partition
nu Bool -> Bool -> Bool
|| Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!Int
0 Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 Bool -> Bool -> Bool
|| Int
m Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = a
forall a. C a => a
one
| Partition -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length Partition
nu Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
m Bool -> Bool -> Bool
&& Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!Int
m Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0 = a
forall a. C a => a
zero
| Int
m Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
1 = [a] -> a
forall a. C a => [a] -> a
product (Int -> a -> [a]
forall a. Int -> a -> [a]
replicate (Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!Int
0) a
x0)
| Maybe a -> Bool
forall a. Maybe a -> Bool
isJust (Array (Int, Int) (Maybe a)
arr Array (Int, Int) (Maybe a) -> (Int, Int) -> Maybe a
forall i e. Ix i => Array i e -> i -> e
! (Partition -> Partition -> Int
_N Partition
lambda Partition
nu, Int
m)) = Maybe a -> a
forall a. HasCallStack => Maybe a -> a
fromJust (Maybe a -> a) -> Maybe a -> a
forall a b. (a -> b) -> a -> b
$ Array (Int, Int) (Maybe a)
arr Array (Int, Int) (Maybe a) -> (Int, Int) -> Maybe a
forall i e. Ix i => Array i e -> i -> e
! (Partition -> Partition -> Int
_N Partition
lambda Partition
nu, Int
m)
| Bool
otherwise = a
s
where
s :: a
s = a -> Int -> a
go (Int -> Int -> Partition -> Array (Int, Int) (Maybe a) -> a
sch (Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) Int
1 Partition
nu Array (Int, Int) (Maybe a)
arr) Int
k
go :: a -> Int -> a
go :: a -> Int -> a
go !a
ss Int
ii
| Partition -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length Partition
nu Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
< Int
ii Bool -> Bool -> Bool
|| Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!(Int
iiInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0 = a
ss
| Bool
otherwise =
let u :: Int
u = Partition
nuPartition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!!(Int
iiInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) in
if Partition -> Int
forall a. [a] -> Int
forall (t :: * -> *) a. Foldable t => t a -> Int
length Partition
nu Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
ii Bool -> Bool -> Bool
&& Int
u Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
0 Bool -> Bool -> Bool
|| Int
u Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Partition
nu Partition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!! Int
ii
then
let nu' :: Partition
nu' = (Int -> IndexedTraversal' Int Partition Int
forall (t :: * -> *) a.
Traversable t =>
Int -> IndexedTraversal' Int (t a) a
element (Int
iiInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) ((Int -> Identity Int) -> Partition -> Identity Partition)
-> Int -> Partition -> Partition
forall s t a b. ASetter s t a b -> b -> s -> t
.~ Int
uInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) Partition
nu in
if Int
u Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
> Int
1
then
a -> Int -> a
go (a
ss a -> a -> a
forall a. C a => a -> a -> a
+ [a]
x[a] -> Int -> a
forall a. HasCallStack => [a] -> Int -> a
!!(Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) a -> a -> a
forall a. C a => a -> a -> a
* Int -> Int -> Partition -> Array (Int, Int) (Maybe a) -> a
sch Int
m Int
ii Partition
nu' Array (Int, Int) (Maybe a)
arr) (Int
ii Int -> Int -> Int
forall a. C a => a -> a -> a
+ Int
1)
else
if Partition
nu' Partition -> Int -> Int
forall a. HasCallStack => [a] -> Int -> a
!! Int
0 Int -> Int -> Bool
forall a. Eq a => a -> a -> Bool
== Int
0
then
a -> Int -> a
go (a
ss a -> a -> a
forall a. C a => a -> a -> a
+ [a]
x[a] -> Int -> a
forall a. HasCallStack => [a] -> Int -> a
!!(Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1)) (Int
ii Int -> Int -> Int
forall a. C a => a -> a -> a
+ Int
1)
else
let arr' :: Array (Int, Int) (Maybe a)
arr' = Array (Int, Int) (Maybe a)
arr Array (Int, Int) (Maybe a)
-> [((Int, Int), Maybe a)] -> Array (Int, Int) (Maybe a)
forall i e. Ix i => Array i e -> [(i, e)] -> Array i e
// [((Partition -> Partition -> Int
_N Partition
lambda Partition
nu, Int
m), a -> Maybe a
forall a. a -> Maybe a
Just a
ss)] in
a -> Int -> a
go (a
ss a -> a -> a
forall a. C a => a -> a -> a
+ [a]
x[a] -> Int -> a
forall a. HasCallStack => [a] -> Int -> a
!!(Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) a -> a -> a
forall a. C a => a -> a -> a
* Int -> Int -> Partition -> Array (Int, Int) (Maybe a) -> a
sch (Int
mInt -> Int -> Int
forall a. C a => a -> a -> a
-Int
1) Int
1 Partition
nu' Array (Int, Int) (Maybe a)
arr') (Int
ii Int -> Int -> Int
forall a. C a => a -> a -> a
+ Int
1)
else
a -> Int -> a
go a
ss (Int
ii Int -> Int -> Int
forall a. C a => a -> a -> a
+ Int
1)
skewSchur'
:: [Rational]
-> Partition
-> Partition
-> Rational
skewSchur' :: [Rational] -> Partition -> Partition -> Rational
skewSchur' = [Rational] -> Partition -> Partition -> Rational
forall a. C a => [a] -> Partition -> Partition -> a
skewSchur
skewSchur :: forall a. AlgRing.C a
=> [a]
-> Partition
-> Partition
-> a
skewSchur :: forall a. C a => [a] -> Partition -> Partition -> a
skewSchur [a]
xs Partition
lambda Partition
mu =
if Partition -> Partition -> Bool
isSkewPartition Partition
lambda Partition
mu
then (a -> Partition -> Int -> a) -> a -> Map Partition Int -> a
forall a k b. (a -> k -> b -> a) -> a -> Map k b -> a
DM.foldlWithKey' a -> Partition -> Int -> a
f a
forall a. C a => a
zero Map Partition Int
lrCoefficients
else [Char] -> a
forall a. HasCallStack => [Char] -> a
error [Char]
"skewSchur: invalid skew partition"
where
lrCoefficients :: Map Partition Int
lrCoefficients = Partition -> Partition -> Map Partition Int
skewSchurLRCoefficients Partition
lambda Partition
mu
f :: a -> Partition -> Int -> a
f :: a -> Partition -> Int -> a
f a
x Partition
nu Int
k = a
x a -> a -> a
forall a. C a => a -> a -> a
+ (Int -> a
forall a. C a => Int -> a
_fromInt Int
k) a -> a -> a
forall a. C a => a -> a -> a
* ([a] -> Partition -> a
forall a. C a => [a] -> Partition -> a
schur [a]
xs Partition
nu)