Matrix representation of (products/duals of) Weyl chambers

weyl_matrix computes a matrix representation of the (polars of) Weyl chambers of finite reflection groups of type A, BC, and D.

weyl_matrix(d, cone_type, product = FALSE)

Arguments

d	vector of dimensions; must be same length as cone_type.
cone_type	vector of cone types; must be same length as d, the available types are as follows: "A": chamber of type A "A_red": chamber of type A, reduced form "BC": chamber of type BC "D": chamber of type D "Ap": polar of chamber of type A "Ap_red": polar of chamber of type A, reduced form "BCp": polar of chamber of type BC "Dp": polar of chamber of type D
product	logical; if TRUE, a representation of the product cone is returned.

Value

If length(d)==1 or (length(d)>1 & product==TRUE) then a single matrix will be returned. If (length(d)>1 & product==FALSE) then a list of matrices will be returned.

Details

The dimensions and types are given in the vectors d and cone_type (vectors must be of same lengths, entries of conetype must be 'A', 'BC', 'D', 'Ap', 'BCp', or 'Dp'). If the length of the vectors is one, a single matrix is returned; if the length of the vectors is greater than one and product==FALSE, a list of matrices is returned; if the length of the vectors is greater than one and product==TRUE, a single matrix representing the product cone is returned.

See also

weyl_ivols

Package: conivol

Examples

weyl_matrix(5, "BC")
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]   -1    1    0    0    0
#> [2,]    0   -1    1    0    0
#> [3,]    0    0   -1    1    0
#> [4,]    0    0    0   -1    1
#> [5,]    0    0    0    0   -1
weyl_matrix(c(5,5), c("BC","Ap"))
#> [[1]]
#>      [,1] [,2] [,3] [,4] [,5]
#> [1,]   -1    1    0    0    0
#> [2,]    0   -1    1    0    0
#> [3,]    0    0   -1    1    0
#> [4,]    0    0    0   -1    1
#> [5,]    0    0    0    0   -1
#> 
#> [[2]]
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
#> [1,]   -1    1   -5   -4   -3   -2   -1
#> [2,]   -1    1    1   -4   -3   -2   -1
#> [3,]   -1    1    1    2   -3   -2   -1
#> [4,]   -1    1    1    2    3   -2   -1
#> [5,]   -1    1    1    2    3    4   -1
#> [6,]   -1    1    1    2    3    4    5
#> 
weyl_matrix(c(5,5), c("BC","Ap"), product = TRUE)
#>       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
#>  [1,]   -1    1    0    0    0    0    0    0    0     0     0     0
#>  [2,]    0   -1    1    0    0    0    0    0    0     0     0     0
#>  [3,]    0    0   -1    1    0    0    0    0    0     0     0     0
#>  [4,]    0    0    0   -1    1    0    0    0    0     0     0     0
#>  [5,]    0    0    0    0   -1    0    0    0    0     0     0     0
#>  [6,]    0    0    0    0    0   -1    1   -5   -4    -3    -2    -1
#>  [7,]    0    0    0    0    0   -1    1    1   -4    -3    -2    -1
#>  [8,]    0    0    0    0    0   -1    1    1    2    -3    -2    -1
#>  [9,]    0    0    0    0    0   -1    1    1    2     3    -2    -1
#> [10,]    0    0    0    0    0   -1    1    1    2     3     4    -1
#> [11,]    0    0    0    0    0   -1    1    1    2     3     4     5

# testing the reduced cones
d <- 6
A <- weyl_matrix(d, "A")
A_red <- weyl_matrix(d, "A_red")
t(A) %*% A
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]    2   -1    0    0    0    0
#> [2,]   -1    2   -1    0    0    0
#> [3,]    0   -1    2   -1    0    0
#> [4,]    0    0   -1    2   -1    0
#> [5,]    0    0    0   -1    2   -1
#> [6,]    0    0    0    0   -1    2
round(t(A_red) %*% A_red, digits=14)
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]    2   -1    0    0    0    0
#> [2,]   -1    2   -1    0    0    0
#> [3,]    0   -1    2   -1    0    0
#> [4,]    0    0   -1    2   -1    0
#> [5,]    0    0    0   -1    2   -1
#> [6,]    0    0    0    0   -1    2

Ap <- weyl_matrix(d, "Ap")
Ap_red <- weyl_matrix(d, "Ap_red")
t(Ap[,-c(1,2)]) %*% Ap[,-c(1,2)]
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]   42   35   28   21   14    7
#> [2,]   35   70   56   42   28   14
#> [3,]   28   56   84   63   42   21
#> [4,]   21   42   63   84   56   28
#> [5,]   14   28   42   56   70   35
#> [6,]    7   14   21   28   35   42
t(Ap_red) %*% Ap_red
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,]   42   35   28   21   14    7
#> [2,]   35   70   56   42   28   14
#> [3,]   28   56   84   63   42   21
#> [4,]   21   42   63   84   56   28
#> [5,]   14   28   42   56   70   35
#> [6,]    7   14   21   28   35   42