SYMF::SfPlethysm
--
plethysm of symmetric functions
SYMF::SfPlethysm(sf1,sf2 <,b1> <,b2>)
sf1, sf2 | - | any symmetric functions |
b1, b2 | - | any known bases |
The SYMF::SfPlethysm
function computes the plethysm sf1[sf2]
.
One may specify that sf1
and sf2
are expressed on
the bases b1
and b2
by using either
, SYMF::SfPlethysm
(sf1,
sf2, b1)
.
SYMF::SfPlethysm
(sf1, sf2, b1, b2)
The plethysm operation is defined as follows: let p[i]
be
the i
-th power-sum symmetric function and sf1
,
sf2
, sf3
be any symmetric functions, then,
(sf1 + sf2)[sf3] = sf1[sf3] + sf2[sf3]
(sf1 x sf2)[sf3] = sf1[sf3] x sf2[sf3]
sf1[p[i]] = p[i][sf1]
p[i][p[j]] = p[i x j]
The default is to compute plethysms on the p
-basis and
return the result expressed on the p
-basis.
Special algorithms have been included to compute plethysms. In particular, the result is expressed on the basis of Schur functions when:
b1=e
, b2=e,h,s
,
b1=h
, b2=e,h,s
,
b1=s
, b2=e,h,s
,
b1=p
, b2=e,h,p,s,m
,
b1=m
, b2=e,h,p,s,m
.
>> muEC::SYMF::SfPlethysm( p[1]*s[2], h[1] );
p[2, 1] p[1, 1, 1] ------- + ---------- 2 2
>> muEC::SYMF::SfPlethysm( e[3], s[2,1], e, s );
s[2, 2, 2, 1, 1, 1] + s[3, 2, 2, 1, 1] + s[3, 3, 1, 1, 1] + s[4, 2, 1, 1, 1] + s[3, 2, 2, 2] + s[3, 3, 2, 1] + s[4, 2, 2, 1] + 2 s[4, 3, 1, 1] + s[5, 2, 1, 1] + s[6, 1, 1, 1] + s[3, 3, 3] + s[4, 3, 2] + s[5, 2, 2] + s[4, 4, 1] + s[5, 3, 1]
MuPAD Combinat, an open source algebraic combinatorics package