towardWeylStandard -- a single Garnir-style straightening step for Weyl fillings

Usage:

towardWeylStandard T
Inputs:
- T, an instance of the type WeylFilling,
Outputs:
- H, a hash table, mapping WeylFillings T' to integer coefficients, with $\sum_{T'} H(T') \cdot T' = T$ in the Weyl module and every key T' strictly larger than T in the term order used by weylStraighten

Description

If T is already Weyl-standard the output is the singleton {T => 1}. Otherwise towardWeylStandard performs one divided-power Garnir shuffle across the first violation (located by isWeylStandard) and returns the resulting linear combination. Repeated application terminates at a $\mathbb{Z}$-linear combination of Weyl-standard fillings; this is exactly what weylStraighten computes.

The replacements are generated by divided-power comultiplication (see divComult) followed by divided-power multiplication (see divMult); the coefficients are products of binomial coefficients, so in general towardWeylStandard produces many terms unlike the sign-flip shuffle on the Schur side.

i1 : T = weyl {{1,2},{0,0}}

     +-+-+
o1 = |1|2|
     |0|0|
     +-+-+

o1 : WeylFilling

i2 : isWeylStandard T

o2 = (0, 0)

o2 : Sequence

i3 : a = towardWeylStandard T

               +-+-+
o3 = HashTable{|0|0| => -1}
               |1|2|
               +-+-+
               +-+-+
               |0|1| => -1
               |0|2|
               +-+-+
               +-+-+
               |0|2| => -1
               |0|1|
               +-+-+

o3 : HashTable

i4 : scan(pairs a, (U, c) -> << "  " << c << " * " << toList U << endl)
  -1 * {{0, 2}, {0, 1}}
  -1 * {{0, 1}, {0, 2}}
  -1 * {{0, 0}, {1, 2}}

For the programmer

The object towardWeylStandard is a function closure.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:1746:0.

towardWeylStandard -- a single Garnir-style straightening step for Weyl fillings

Description

See also

For the programmer