dualLR -- GL-equivariant projection S_nu V ⊗ S_mu V --> S_lambda V at a chosen LR tableau

Usage:

dualLR((lambda, mu, nu), Q, n)

dualLR((lambda, mu, nu), Q, P)

dualLR(..., Convention => "Row" | "Filling" | "Weyl")
Inputs:
- (lambda, mu, nu), a sequence, three partitions with $|\lambda| = |\mu| + |\nu|$ and $S_\lambda V$ appearing in $S_\nu V \otimes S_\mu V$ (i.e. $c^\lambda_{\mu,\nu} > 0$)
- Q, a list, an LR tableau of shape $\lambda/\mu$ with content $\nu$ (as returned by lrTableaux) singling out one of the $c^\lambda_{\mu,\nu}$ copies of $S_\lambda V$
- n, an integer, or pass a polynomial ring whose numgens is used
Optional inputs:
- Convention => a string, default value "Row", , basis convention for the three Schur factors.
Outputs:
- a matrix, a $\mathbb{Q}$-matrix of size $\dim S_\lambda V \times (\dim S_\nu V \cdot \dim S_\mu V)$, the $GL_n$-equivariant projection onto Q's specific copy of $S_\lambda V$ inside $S_\nu V \otimes S_\mu V$

Description

Returns the $GL_n$-equivariant projection onto the LR-summand of $S_\nu V \otimes S_\mu V$ indexed by Q (unique up to scalar by Schur's lemma). When the LR coefficient $c^\lambda_{\mu,\nu}$ exceeds 1, distinct LR tableaux Q give different projections, each onto its own copy of $S_\lambda V$; $\mathrm{dualLR}_Q \circ \mathrm{lrMap}_{Q'} = 0$ when $Q \neq Q'$, and $\mathrm{dualLR}_Q \circ \mathrm{lrMap}_Q = \mathrm{Id}$.

Construction: stack the LR inclusions lrMap for every $(\lambda', Q')$ with $c^{\lambda'}_{\mu,\nu} > 0$ into a square invertible matrix on $S_\nu V \otimes S_\mu V$ (square by the LR/Pieri rule) and read off the rows of its inverse corresponding to $(\lambda, Q)$.

i1 : shapes = ({2,1}, {1}, {1,1});

i2 : Q = (lrTableaux shapes)#0;

i3 : M = lrMap(shapes, Q, 3);   -- inclusion S_(2,1) V -> S_(1,1) V ⊗ S_(1) V

              9       8
o3 : Matrix QQ  <-- QQ

i4 : N = dualLR(shapes, Q, 3);  -- projection back onto Q's copy

              8       9
o4 : Matrix QQ  <-- QQ

i5 : N * M == id_(QQ^(numColumns M))

o5 = true

i6 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1});  -- multiplicity-2 case

i7 : M0 = lrMap(({3,2,1},{2,1},{2,1}), Qs#0, 3);

              64       8
o7 : Matrix QQ   <-- QQ

i8 : M1 = lrMap(({3,2,1},{2,1},{2,1}), Qs#1, 3);

              64       8
o8 : Matrix QQ   <-- QQ

i9 : N0 = dualLR(({3,2,1},{2,1},{2,1}), Qs#0, 3);

              8       64
o9 : Matrix QQ  <-- QQ

i10 : N0 * M0 == id_(QQ^(numColumns M0))   -- recovers Q_0's copy

o10 = true

i11 : N0 * M1 == 0                          -- annihilates Q_1's copy

o11 = true

Ways to use dualLR:

dualLR(Sequence,List,PolynomialRing)
dualLR(Sequence,List,ZZ)

For the programmer

The object dualLR is a method function with options.

The source of this document is in PieriMaps/doc.m2:1133:0.

dualLR -- GL-equivariant projection S_nu V ⊗ S_mu V --> S_lambda V at a chosen LR tableau

Description

See also

Ways to use dualLR:

For the programmer