Errata and Addenda

“A Combinatorial Description of the Syzygies of Certain Weyl Modules” by Mari Sano *, Communications in Algebra , 31(10), pp. 5117–5167, 2003.

The paper mentioned constructs a basis for the syzygies for the Buchsbaum–Rota resolution

of certain Weyl modules K _λ/μ (where λ/μ = (p + t ₁ + t ₂, q + t ₂, r)/(t ₁ + t ₂, t ₂, 0)), associated to the skew-shape

where the number of triple overlaps is at most 1, i.e., r − t ₁ − t ₂ ≤ 1.

The syzygies are constructed via a partition of the canonical Letter–Place basis of bistandard bitableaux of each module P _i in complementary subsets, the essential and non-essential elements, the syzygies are given as the images under the boundary operator of the essential elements.

A crucial point of the construction is the so-called Rank condition, that says that the number of essential elements in P _i+1 is the same as the number of non-essential elements in P _i . This is proven by a Littlewood–Richardson argument for i = 0, (Theorem 2, p. 5133) which is correct, and by a direct correspondence for i > 0 (Theorem 4, p. 5153 for i = 1 and Theorem 6, p. 5160 for i > 1). This correspondence is not correct for all skew-shapes under consideration. The reason is that some essential bistandard bitableaux might be sent by one of the cases in the correspondence to non-essential elements that are not bistandard (or, form another viewpoint, certain bistandard essential elements are not reached under the correspondence as images of bistandard non-essential elements).

Let us give an example: consider the shape

The element

is a bistandard essential bitableaux, and under the correspondence is sent to

which is non-standard.

Note that

Thus the boundary map applied to this element contains non-standard terms (namely, the first term); therefore for this shape, in order to compute the boundary map in terms of the canonical basis a straightening algorithm must be used.

Let us now describe the set of 3-rowed skew shapes such that the constructions in Sano (2003) do work: let s be the number of overlaps between the second and third rows (thus s + t ₂ = r). We have

Lemma 1

The correspondences in Theorems 4 and 6 of Sano (2003) hold true for skew shapes

such that q − p ≥ s − t ₂ − 1 holds.

Proof

We do the general case (Theorem 6; Theorem 4 is a particular case which was written separately for presentation purposes); in that theorem there is only a case of the correspondence that is incorrect, namely, the non-essential elements of the form

where t ₁ ≥ σ₁ ≥ q − p, β₁ ≥ t ₂ + 1, β _j  > 0 for j = 2,…, i, , ρ₁ + ρ₂ + ρ₃ = r − |β| and ρ₁ > 0 correspond to essential elements of the form

We will show that under the hypothesis q − p ≥ s − t ₂ − 1, this is a bijection of bistandard bitableaux of the given forms.

It is clear that given a non-essential bistandard bitableaux element of the form (1), the corresponding element of the form (2) is an essential bistandard bitableaux.

Let us prove the other way around, that is, given a essential bistandard bitableaux element of the form (2) the corresponding element of the form (1) is a non-essential bistandard bitableaux. All we need to show is that p + σ₁ + |β| ≥ q − σ₁ + ρ₂.

On the one hand,

where |β| = t ₂ + 1 + β'. So

On the other hand,

because q − p ≥ s − t ₂ − 1.

Therefore the results of Sano (2003) hold for these kind of shapes.

Let us compare the scope of Sano (2003) with what was actually proven; in Sano (2003), the results were supposed to hold for 3-rowed Weyl modules with at most one triple overlap. This erratum says that we can only assert the result for the 3-rowed skew shapes that, in addition, satisfy the relation (R) = q − p ≥ s − t ₂ − 1. In order to get an idea of how many they are, we have the following easy to prove fact:

Lemma 2

Let S be a 3-rowed skew shape that does not satisfy (R). Then the shape S′ obtained by rotating S 180 degrees satisfies (R).

Thus the shapes satisfying (R) are in a sense, more than half of the shapes considered (some shapes satisfying (R) still satisfy (R) after being rotated 180 degrees).

It is interesting to note that this condition implies that the boundary map takes basis elements into linear combinations of basis elements directly, i.e., no straightening is necessary.

Also, under the stronger condition q − p ≥ s − 1, a splitting contracting homotopy can be constructed (Sano, 2004).

Acknowledgments

Notes

^#Communicated by W. Bruns.

*Correspondence: Mari Sano, USA; Email: [email protected].