standardTableaux -- all semistandard Young tableaux of a given shape

Usage:

standardTableaux(d, mu)
Inputs:
- d, an integer, the size of the alphabet $\{0,\ldots,d-1\}$
- mu, a list, a partition giving the column lengths of the Filling (equivalently, the conjugate of the row shape $\lambda$; see schurModule)
Outputs:
- a list, all semistandard Fillings with column lengths $\mu$ and entries in $\{0,\ldots,d-1\}$

Description

The output is a $\mathbb{Z}$-basis of schurModule(mu, R^d); its cardinality equals the Weyl dimension $\dim S_\mu(\mathbb{Q}^d)$, given by the hook-content formula $$\dim S_\mu(\mathbb{Q}^d) = \prod_{(i,j)\in\mu} \frac{d - i + j}{h(i,j)},$$ where $h(i,j)$ is the hook length at cell $(i,j)$.

i1 : S = standardTableaux(3, {2,1});

i2 : #S

o2 = 8

i3 : rank schurModule({2,1}, QQ^3)

o3 = 8

For $\mu = (2,1)$ and $d = 3$ the hook-content formula gives $(3 \cdot 4 \cdot 2)/(3 \cdot 1 \cdot 1) = 8$.

i4 : standardTableaux(2, {1,1})

      +-+-+  +-+-+  +-+-+
o4 = {|0|0|, |0|1|, |1|1|}
      +-+-+  +-+-+  +-+-+

o4 : List

i5 : standardTableaux(4, {3})

      +-+  +-+  +-+  +-+
o5 = {|0|, |0|, |0|, |1|}
      |1|  |1|  |2|  |2|
      |2|  |3|  |3|  |3|
      +-+  +-+  +-+  +-+

o5 : List

Ways to use standardTableaux:

standardTableaux(ZZ,List)

For the programmer

The object standardTableaux is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:1488:0.

standardTableaux -- all semistandard Young tableaux of a given shape

Description

See also

Ways to use standardTableaux:

For the programmer