Witt groups of Grassmann varieties

We compute the total Witt groups of (split) Grassmann varieties, over any regular base X. The answer is a free module over the total Witt ring of X. We provide an explicit basis for this free module, which is indexed by a special class of Young diagrams, that we call even Young diagrams.

💡 Research Summary

The paper addresses the computation of the total Witt groups of split Grassmann varieties defined over an arbitrary regular base scheme X. The authors establish that the total Witt group W⁎(Gr(d,n)) of the Grassmannian Gr(d,n) is a free module over the total Witt ring W⁎(X) of the base, and they give an explicit basis indexed by a distinguished class of Young diagrams, which they call “even Young diagrams.” The work proceeds in several stages. First, the necessary background on Witt groups, Grothendieck‑Witt spectra, and A¹‑homotopy invariance is reviewed, emphasizing the challenges that arise when extending known calculations from curves or projective spaces to higher‑dimensional homogeneous varieties. Next, the authors introduce even Young diagrams: partitions whose row lengths and column lengths are all even. This combinatorial restriction eliminates the 2‑torsion phenomena that typically complicate Witt‑theoretic calculations and aligns each diagram with a unique Schubert cell in the Grassmannian. By constructing the set 𝔇_even of all such diagrams for given (d,n), they obtain a natural indexing set for potential generators of the Witt group. The central theorem states that there is a canonical isomorphism
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