Let \( \mathbb{F} \) be the finite field of q elements and let \( \mathcal{P}\left( {\mathrm{n},\mathrm{q}} \right) \) denote the projective space of dimension n − 1 over \( \mathbb{F} \). We construct a family \( \mathrm{H}_{\mathrm{k},\mathrm{i}}^{\mathrm{n}} \) of combinatorial homology modules associated to \( \mathcal{P}\left( {\mathrm{n},\mathrm{q}} \right) \) for coefficient fields of positive characteristic co-prime to q. As FGL(n, q)-representations these modules are obtained from the permutation action of GL(n, q) on the Grassmannians of \( {{\mathbb{F}}^{\mathrm{n}}} \). We prove a branching rule for \( \mathrm{H}_{\mathrm{k},\mathrm{i}}^{\mathrm{n}} \) and use this to determine the homology representations completely. Our results include a duality theorem and the characterization of \( \mathrm{H}_{\mathrm{k},\mathrm{i}}^{\mathrm{n}} \) through the standard irreducibles of GL(n, q) over F.
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 414, 2013, pp. 156–180.
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Siemons, J., Smith, D. Some Homology Representations for Grassmannians in Cross-Characteristics. J Math Sci 199, 329–342 (2014). https://doi.org/10.1007/s10958-014-1861-8
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DOI: https://doi.org/10.1007/s10958-014-1861-8