When a symmetrizer sandwich vanishes #
For a μ-tableau t with row symmetrizer a_t and column antisymmetrizer b_t, the
irreducibility of the left ideal ℚ[Sₙ] c_t generated by the Young symmetrizer
c_t = a_t b_t is driven by the behaviour of the sandwiches a_t σ b_t, for σ a permutation
of the labels. This file proves the criterion that governs them.
The criterion is a row/column intersection condition, recorded here as
YoungTableau.RowMeetsColumnTwice t s: two distinct labels lie in a common row of t and in a
common column of s, so that row and that column share not just one label but two. The
key vanishing lemma
YoungTableau.RowMeetsColumnTwice.rowSymmetrizer_mul_single_mul_columnAntisymmetrizer_eq_zero
says that a_t σ b_t = 0 as soon as a row of t meets a column of the relabelled tableau
relabel σ t in two distinct labels. Its proof is the transposition trick: the swap τ of the
two labels lies in the row group, its conjugate σ⁻¹ τ σ lies in the column group, and pushing
τ across σ turns the sandwich into its own negative.
The direction of σ matters, and it is the one written here: the two labels sharing a column are
σ⁻¹ x and σ⁻¹ y, or equivalently x and y share a column of the relabelling σt, which is
Fulton's formulation. The variant asking instead that σ x and σ y share a column of t is
false, as the sandwich lemmas below already show: if x ≠ y share a row of t and x ≠ v share
a column of t, then σ = (x y) (x v) carries x and y to the common-column pair v and x,
yet σ lies in Row(t) · Col(t), so a_t σ b_t ≠ 0 by
YoungTableau.rowSymmetrizer_mul_single_mul_columnAntisymmetrizer_ne_zero.
In the opposite direction, a permutation of the form σ = p q with p in the row group and q
in the column group has a_t σ b_t = sign q • c_t ≠ 0
(YoungTableau.rowSymmetrizer_mul_single_mul_columnAntisymmetrizer_eq_sign_smul_youngSymmetrizer),
and no row of t meets a column of relabel σ t twice
(YoungTableau.not_rowMeetsColumnTwice_relabel_mul). So the criterion never fires on the
permutations whose sandwich is visibly nonzero, and it certifies that the permutations it does
fire on lie outside the product set Row(t) · Col(t).
A relabelling can be moved from one argument of the criterion to the other by inverting it
(YoungTableau.rowMeetsColumnTwice_relabel_left_iff), which is what lets the criterion be applied
when it is the tableau whose rows are read that has been relabelled.
Neither half is vacuous: YoungTableau.exists_rowMeetsColumnTwice_relabel produces a permutation
the criterion fires on whenever t has a row and a column each containing two distinct labels, so
YoungTableau.exists_rowSymmetrizer_mul_single_mul_columnAntisymmetrizer_eq_zero exhibits a
vanishing sandwich for every shape other than a single row or a single column.
Note that the criterion, and not the cruder condition σ ∉ Row(t) · Col(t), is what the
transposition trick establishes; the converse implication (a permutation on which the criterion
fails already factors as p q) is a separate combinatorial theorem and is not proved here.
References #
- W. Fulton, Young Tableaux, Section 7.2.
- W. Fulton and J. Harris, Representation Theory: A First Course (1991), Lecture 4.
- Schur--Weyl roadmap, Layer 2.
A row of the tableau t meets a column of the tableau s twice if two distinct
labels lie in a common row of t and in a common column of s. A row of t and a column of t
share at most one label, so meeting once is no condition at all and it is the second shared label
that carries the content. The relation is not symmetric: rows are read off t and columns off
s.
Equations
Instances For
The defining characterisation of RowMeetsColumnTwice, for introducing and eliminating the
predicate on arbitrary tableaux. Not a simp lemma: it would shadow the normal forms
rowMeetsColumnTwice_relabel_iff and not_rowMeetsColumnTwice_self.
A row of t meets a column of the relabelling relabel σ t exactly when two distinct labels
of a common row of t have their σ-preimages in a common column of t.
Relabelling moves across the criterion by inverting: a row of relabel σ t meets a column of
s twice exactly when a row of t meets a column of relabel σ⁻¹ s twice. Both sides say that
two distinct labels share a row of t after applying σ⁻¹ and share a column of s, read from
the two ends.
Not a simp lemma: neither side is simpler than the other, and pushing the relabelling to the
right would fight rowMeetsColumnTwice_relabel_iff.
No row of a tableau meets one of its own columns twice: a label is determined by its row together with its column.
The criterion is satisfiable as soon as t has a row containing two distinct labels and a
column containing two distinct labels: some relabelling of t then has a column meeting that row
twice. Concretely one moves x and y onto u and v and relabels by the inverse.
The key vanishing lemma, in terms of explicit labels: if the distinct labels x and y
lie in a common row of t while their σ-preimages lie in a common column of t, then the
sandwich a_t σ b_t vanishes.
The key vanishing lemma. If a row of t meets a column of the relabelled tableau
relabel σ t in two distinct labels, then the sandwich a_t σ b_t vanishes.
A permutation lying in the product of the row and the column group has its sandwich equal to the Young symmetrizer, up to the sign of its column part.
A permutation lying in the product of the row and the column group has a nonzero sandwich.
No row of t meets a column of relabel (p * q) t twice, when p lies in the row group and
q in the column group: the two labels would share a row and a column of t itself.
A permutation on which the row/column criterion fires lies outside the product set
Row(t) · Col(t).
The vanishing lemma is not vacuous: a tableau with a row containing two distinct labels and a column containing two distinct labels admits a permutation whose sandwich vanishes.
The contrapositive of the key vanishing lemma: a nonzero sandwich forces every two distinct
labels of a common row of t into distinct columns of relabel σ t.