SP::TablePe
--
table of the Pe-basis
SP::TablePe(n)
n | - | any positive integer denoting the degree of a symmetric group |
The SP::TablePe
function returns the table of all elements of the Pe
-basis,
which is the dual basis of the basis of (increasing) monomials (less than
the ``staircase'' monomial), i.e. the basis of products of elementary
symmetric functions on an alphabet flag.
For instance, Pe[1, 2, 0, 1, 0]
stands for the product
e1(x1,x2,x3,x4)*e2(x1,x2,x3)*e1(x1)
.
When called with the second argument X
, it returns the table of
all polynomials in Sn, expressed in the Schubert basis.
>> t:=muEC::SP::TablePe(3);
table( 2 [2, 1, 0] = x1 x2, 2 [1, 1, 0] = x1 x2 + x1 , [1, 0, 0] = x1 + x2, [2, 0, 0] = x1 x2, [0, 1, 0] = x1, [0, 0, 0] = 1 )
>> muEC::SP::TablePe(3, X);
table( [2, 1, 0] = X[3, 2, 1], [1, 1, 0] = X[2, 3, 1] + X[3, 1, 2], [1, 0, 0] = X[1, 3, 2], [2, 0, 0] = X[2, 3, 1], [0, 1, 0] = X[2, 1], [0, 0, 0] = X[1] )
MuPAD Combinat, an open source algebraic combinatorics package