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Some homology representations for Grassmannians in cross-characteristics
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 156-180 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\mathbb F$ be the finite field of $q$ elements and let $\mathcal P(n,q)$ denote the projective space of dimension $n-1$ over $\mathbb F$. We construct a family $H^n_{k,i}$ of combinatorial homology modules associated to $\mathcal P(n,q)$ for coefficient fields of positive characteristic co-prime to $q$. As $F\mathrm{GL}(n,q)$-representations these modules are obtained from the permutation action of $\mathrm{GL}(n,q)$ on the Grassmannians of $\mathbb F^n$. We prove a branching rule for $H^n_{k,i}$ and use this to determine the homology representations completely. Our results include a duality theorem and the characterisation of $H^n_{k,i}$ through the standard irreducibles of $\mathrm{GL}(n,q)$ over $F$. 
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     title = {Some homology representations for {Grassmannians} in cross-characteristics},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_414_a9/}
}
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J. Siemons; D. Smith. Some homology representations for Grassmannians in cross-characteristics. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 25, Tome 414 (2013), pp. 156-180. http://geodesic.mathdoc.fr/item/ZNSL_2013_414_a9/

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