Reading order, the superstandard tableaux, and the shapes with a unique standard tableau #
Numbering the cells of a Young diagram μ in reading order -- left to right along the first
row, then left to right along the second, and so on -- labels the cell (i, j) by
YoungDiagram.readingIndex μ (i, j) = μ.rowLen 0 + ⋯ + μ.rowLen (i - 1) + j. Reading
order increases along rows and down columns, so this labelling is a standard Young tableau, the
row-superstandard tableau TauCeti.StandardYoungTableau.rowSuperstandard μ; in particular
TauCeti.standardCount μ, the number f^μ of standard Young tableaux, is never zero.
Numbering the cells column by column instead gives a second standard Young tableau
TauCeti.StandardYoungTableau.colSuperstandard μ, the transpose of the row-superstandard tableau
of the transposed diagram. The two disagree as soon as μ has a cell outside its first row and
a cell outside its first column, while a diagram with at most one row, or with at most one column
-- the empty diagram included -- has only one standard Young tableau at all. So f^μ = 1 holds
exactly for the diagrams with at most one row and those with at most one column.
Main definitions #
YoungDiagram.readingIndex: the position of a cell in reading order.TauCeti.StandardYoungTableau.rowSuperstandard: the tableau numbering the cells row by row.TauCeti.StandardYoungTableau.colSuperstandard: the tableau numbering the cells column by column.
Main results #
TauCeti.standardCount_pos: every Young diagram has a standard Young tableau.TauCeti.standardCount_eq_one_iff: a Young diagram has exactly one standard Young tableau precisely when it has at most one row or at most one column, the empty diagram included.
References #
- W. Fulton, Young Tableaux, Section 1.1.
- Schur--Weyl roadmap, Layers 0 and 5.
The position of the coordinate pair c in reading order: the number of cells of μ lying in
rows above c.1, plus the column index c.2. This is defined for an arbitrary pair, but it
numbers μ in reading order only on the cells of μ.
Equations
- μ.readingIndex c = ∑ i ∈ Finset.range c.1, μ.rowLen i + c.2
Instances For
The reading index of a coordinate pair is the sum of the lengths of the rows of μ above it,
plus its column index.
In row 0 the reading index is the column index; for a cell of the first row of μ this says
that it is numbered by its column index.
The reading index of (1, 0) is the length of the first row of μ; when μ has a second row,
(1, 0) is its first cell.
The first k row lengths count the cells in the first k rows, so the reading index of a
cell of μ is smaller than the number of cells of μ.
Reading order runs down the rows: a cell in an earlier row has the smaller reading index.
Reading order runs along each row: within a row, a cell in an earlier column has the smaller reading index.
Distinct cells of μ have distinct reading indices.
The row-superstandard tableau of μ: the standard Young tableau numbering the cells of
μ in reading order, left to right along the first row, then left to right along the second, and
so on. This is the tableau at which the classical-groups roadmap fixes the Young symmetrizer
c_λ.
Equations
- TauCeti.StandardYoungTableau.rowSuperstandard μ = { toTableau := Equiv.ofBijective (fun (c : ↥μ.cells) => ⟨μ.readingIndex ↑c, ⋯⟩) ⋯, row_strict' := ⋯, col_strict' := ⋯ }
Instances For
The row-superstandard tableau labels a cell by its reading index.
The column-superstandard tableau of μ: the standard Young tableau numbering the cells of
μ top to bottom down the first column, then top to bottom down the second, and so on. It is the
transpose of the row-superstandard tableau of the transposed diagram.
Equations
Instances For
The column-superstandard tableau labels a cell by the reading index of the transposed cell in the transposed diagram.
Transposing the column-superstandard tableau of μ gives the row-superstandard tableau of the
transposed diagram: this is the defining property of colSuperstandard.
Every Young diagram carries a standard Young tableau.
Diagrams with a unique standard Young tableau #
A diagram with at most one row admits only the row-superstandard tableau: a standard Young tableau on a single row labels the cells in increasing order of column index.
A diagram with at most one row has at most one standard Young tableau.
A diagram with at most one column has at most one standard Young tableau.
If μ has a cell outside its first row and a cell outside its first column, then the
row-superstandard and the column-superstandard tableaux differ: reading by rows labels the cell
(0, 1) by 1, while reading by columns labels it by the length of the first column.
Counting standard Young tableaux of extreme shapes #
Every Young diagram has a standard Young tableau, namely the row-superstandard one.
The number of standard Young tableaux of a given shape is never zero.
A Young diagram with at most one row has exactly one standard Young tableau.
A Young diagram with at most one column has exactly one standard Young tableau.
A Young diagram with a cell outside its first row and a cell outside its first column has at least two standard Young tableaux.
The shapes with a unique standard Young tableau are exactly the diagrams with at most one row and those with at most one column; the empty diagram, which satisfies both hypotheses, is one of them.