The Specht module of the shape (N-1, 1) is the standard representation #
Throughout, N is the degree of the symmetric group, and n the parameter the declarations for
the concrete partition carry, so that N = n + 2.
The third of the named small irreducible representations of S_N, beside the trivial
representation S^{(N)} and the sign representation S^{(1^N)} of
TauCeti.RepresentationTheory.Symmetric.Specht.Extremes, is the (N-1)-dimensional standard
representation S^{(N-1,1)}. Unlike those two it is not a line, and identifying it needs the
tabloid combinatorics of a shape with two rows. This file supplies that combinatorics and proves
the identification.
The shape is pinned by its row lengths: μ.rowLen 1 = 1 says the second row is a single cell, and
μ.rowLen 2 = 0 says there is no third row, so μ is a shape (m, 1); the partition (n+1, 1)
of n+2 is TauCeti.Nat.Partition.singletonSecondRow n, and both hypotheses hold for its diagram.
Everything is stated on the diagram, where the polytabloids live, and specialized to that partition
at the end.
Three facts about such a shape drive the whole file, and each is elementary once the two row
lengths are fixed. The second row holds a single label, TauCeti.YoungTableau.secondRowLabel; the
first column holds exactly two, that one and TauCeti.YoungTableau.firstColumnLabel; and every
other column holds exactly one. Hence the column group of a tableau is
{1, (firstColumnLabel secondRowLabel)} (TauCeti.YoungTableau.mem_colSubgroup_iff), so the
antisymmetrization defining the polytabloid has just two terms and
e_t = {t} - (a b) · {t}
is a difference of two tabloid basis vectors
(TauCeti.YoungTableau.polytabloid_eq_single_sub_single). Differences of basis vectors span
exactly the vectors whose coefficients sum to zero, which is what
TauCeti.augmentationSubrepresentation is; and conversely every such difference is a
polytabloid, because a tableau can be relabelled, without moving the label of its short row, so
that the top of its first column carries any prescribed other label. The Specht module is
therefore the augmentation subrepresentation of the Young permutation module
(TauCeti.spechtSubrepresentation_eq_augmentationSubrepresentation), which is what
TauCeti.standardRepresentation is by definition.
To read that on the labels rather than on the tabloids, the file also names the tabloids of such a
shape by the labels: a tabloid splits the labels into the long row and a single short one, so it is
named by the label of the short row, equivariantly
(TauCeti.labelTabloidEquiv, TauCeti.labelTabloid_smul). Transporting ℚ[Fin μ.card] along
that naming carries the standard representation of Fin μ.card onto the Specht module
(TauCeti.standardRepresentationEquivSpechtSubrepresentation), and the dimension μ.card - 1
follows. Relabelling the labels of the diagram along TauCeti.card_diagramOf states both for the
partition (n+1, 1) itself.
The parallel statement one level up, M^{(N-1,1)} = triv ⊕ standard, is proved for the partition
(n+1, 1) itself in
TauCeti.RepresentationTheory.Symmetric.PermutationModule.SingletonSecondRow; that file names the
tabloids by the labels through the Young subgroup, which is the stabilizer of a point, whereas the
naming here is built from the tableau combinatorics, so that it is available on the diagram the
polytabloids are indexed by.
Main definitions #
TauCeti.YoungTableau.secondRowLabelandTauCeti.YoungTableau.firstColumnLabel: the label of the single cell of the second row, and the label above it.TauCeti.labelTabloidandTauCeti.labelTabloidEquiv: the tabloids of a shape(m, 1)are the labels,TauCeti.labelTabloidRepresentationEquivbeing the induced identification ofM^μwith the permutation module on the labels.TauCeti.standardRepresentationEquivSpechtSubrepresentation:S^{(N-1,1)}is the standard representation, andTauCeti.standardRepresentationEquivSpechtModuleSingletonSecondRowthe same identification for the partition-indexedTauCeti.spechtModule.
Main results #
TauCeti.YoungTableau.mem_colSubgroup_iff: the column group of a tableau of a shape(m, 1)has two elements, andTauCeti.YoungTableau.rowSubgroup_eq_stabilizer: its row group is the stabilizer of the label of the short row.TauCeti.YoungTableau.polytabloid_eq_single_sub_single: a polytabloid of such a shape is a difference of two tabloids, andTauCeti.YoungTableau.tabloid_eq_iff_secondRowLabel_eq: a tabloid is named by the label of its short row.TauCeti.spechtSubrepresentation_eq_augmentationSubrepresentation: the Specht module of a shape(m, 1)is the augmentation subrepresentation ofM^μ.TauCeti.finrank_spechtSubrepresentation_of_rowLenandTauCeti.finrank_spechtModule_singletonSecondRow: its dimension is one less than the number of labels, soS^{(n+1,1)}has dimensionn+1.
References #
- G. D. James, The Representation Theory of the Symmetric Groups, Chapter 5, where
S^{(N-1,1)}is identified with the standard representation. - W. Fulton, Young Tableaux, Section 7.2.
The two cells of the first column #
The label a tableau puts in the single cell of the second row.
Instances For
The label a tableau puts in the top cell of the first column.
Instances For
Relabelling moves the second-row label along.
Relabelling moves the top label of the first column along.
On a shape with no third row every label lies in the first or the second row.
The second row holds exactly one label.
The second-row label and the top label of the first column are distinct: they lie in different rows.
The first column holds exactly two labels.
The row and the column group #
On a shape (m, 1) the row group is the stabilizer of the second-row label. A permutation
keeps every label in its row exactly when it fixes the one label of the short row.
On a shape (m, 1) the column group has two elements: the identity and the transposition
of the two labels of the first column.
The polytabloid of a shape (m, 1) #
The transposition of the two labels of the first column does move the tabloid: the two labels lie in different rows.
The polytabloid of a shape (m, 1) is a difference of two tabloids. The column group of
t consists of the identity and the transposition of the two labels of the first column, so the
antisymmetrization of {t} has just two terms, with opposite signs.
A tabloid of a shape (m, 1) is named by the label of its short row.
The Specht module of a shape (m, 1) #
A polytabloid of a shape (m, 1) has vanishing coefficient sum: it is a difference of two
tabloids.
Every difference of two tabloids of a shape (m, 1) is a polytabloid, hence lies in the
Specht module: a tableau whose short row carries the label naming the first tabloid can be
relabelled, without moving that label, so that the top of its first column carries the label naming
the second.
The Specht module of a shape (m, 1) is the standard representation of the tabloids. The
polytabloids of such a shape are exactly the differences of two tabloid basis vectors, and those
span the subrepresentation on which the coefficients sum to zero.
Since TauCeti.standardRepresentation is by definition the action carried by the augmentation
subrepresentation of a permutation module, this is the identification S^μ = standard,
read on the tabloids; TauCeti.labelTabloidEquiv names the tabloids by the labels.
The tabloids of a shape (m, 1) are the labels #
The tabloid named by a label: the tabloid of a shape (m, 1) whose short row carries the
prescribed label. Relabelling a tableau of the shape by the transposition carrying its own
short-row label to the prescribed one produces it; which tableau is relabelled is irrelevant, since
a tabloid of such a shape is named by the label of its short row, so the naming is canonical
(TauCeti.labelTabloid_eq_tabloid_relabel).
Equations
Instances For
The tabloid named by a label is read off any tableau of the shape: relabel it by the transposition carrying the label of its short row to the prescribed one.
Naming a tabloid by the label of its short row is equivariant. The left-hand side is stated
with σ k rather than the σ • k of the equivariance interfaces, since Equiv.Perm.smul_def
is simp; the two are definitionally equal.
The tabloids of a shape (m, 1) are the labels. A tabloid of such a shape splits the
labels into a long row and a single short one, so it is named by the label of the short row.
Equations
- TauCeti.labelTabloidEquiv h1 h2 = Equiv.ofBijective (TauCeti.labelTabloid h1) ⋯
Instances For
The naming of the tabloids by the labels, read backwards: the label naming a tabloid is the label of the short row of any tableau representing it.
The Young permutation module of a shape (m, 1) is the permutation module on the labels.
Equations
Instances For
The Specht module of a shape (m, 1) is the standard representation. Naming the tabloids
by the labels of their short rows carries ℚ[Fin μ.card] onto the Young permutation module and its
augmentation subrepresentation -- the standard representation -- onto the Specht module, which
TauCeti.spechtSubrepresentation_eq_augmentationSubrepresentation identifies as the augmentation
subrepresentation on the tabloids.
This is the third of the named small irreducibles of Sₙ, beside the trivial representation
S^{(n)} and the sign representation S^{(1ⁿ)} of
TauCeti.RepresentationTheory.Symmetric.Specht.Extremes.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The underlying equivalence of the identification is the transport of the tabloids.
The Specht module of a shape (m, 1) has dimension one less than the number of labels:
the standard representation of the symmetric group on μ.card labels has dimension
μ.card - 1.
The shape (n+1, 1) #
S^{(n+1,1)} has dimension n + 1: the Specht module of the shape (n+1, 1) of n + 2
is the (n+1)-dimensional standard representation of S_{n+2}.
S^{(n+1,1)} is the standard representation, for the partition-indexed Specht module
TauCeti.spechtModule the classification of the irreducibles is stated in. This is
TauCeti.standardRepresentationEquivSpechtSubrepresentation for the shape (n+1, 1), with the
symmetric group on the labels of the diagram identified with S_{n+2} along
TauCeti.card_diagramOf, the relabelling TauCeti.spechtModule is defined by.
Equations
Instances For
The identification of S^{(n+1,1)} with the standard representation is the transport of the
tabloids, read on the labels of the diagram along TauCeti.card_diagramOf.