eVariable -- Elementary symmetric functions in a Symmetric ring

Description

For a Symmetric ring R of dimension n, R.eVariable is a function which assigns to each index 1\leq i\leq n the i-th elementary symmetric function. If i is outside the given bounds, an error is returned.

i1 : R = symmetricRing(QQ,5,EHPVariables => (a,b,c));

i2 : R.eVariable 3

o2 = a
      3

o2 : R

The elementary generators are related to the Schur basis via e_k = s_{1^k}; toS realizes this on any product of e-variables.

i3 : R = symmetricRing(QQ,4);

i4 : toS (R.eVariable 2)

o4 = s
      1,1

o4 : schurRing (QQ, s, 4)

i5 : toS ((R.eVariable 2)^2)

o5 = s    + s      + s
      2,2    2,1,1    1,1,1,1

o5 : schurRing (QQ, s, 4)

One may also use eVariable from a Symmetric ring built with a non-standard coefficient ring or group acting on it.

i6 : Rsn = symmetricRing(QQ,4,GroupActing => "Sn");

i7 : toS symmetricPower(2, Rsn.eVariable 2)

o7 = s
      2

o7 : schurRing (QQ, s, 4)

i8 : Rmod = symmetricRing(ZZ/5,3);

i9 : (Rmod.eVariable 1)^5

      5
o9 = e
      1

o9 : Rmod

For the programmer

The object eVariable is a symbol.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:5355:0.

eVariable -- Elementary symmetric functions in a Symmetric ring

Description

See also

For the programmer