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Given a projective variety acted on by an algebraic torus, we introduce the notion of ( W ,  R )-matroids using the fixed-point set  W  and the set  R  of equivalence classes of one-parameter subgroups. The ( W ,  R )-matroids provide close links among the geometry of torus orbits and Schubert-type cells, the theory of momentum polyhedra, and the combinatorial geometries. On the way to establishing the main theme of the paper, we showed that there are only finitely many Bialynicki-Birula decompositions induced by infinitely many one-parameter subgroups.

$(W,R)$-matroids and thin Schubert-type cells attached to algebraic torus actions