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Schrijver, Alexander
Bases of many weights ★★★
Let be an (additive) abelian group, and for every
let
.
Conjecture Let
be a matroid on
, let
be a map, put
and
. Then
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ w : E \rightarrow G $](/files/tex/7c1a9b6ba2a67b002e25339803f3d5a6da1b684d.png)
![$ S = \{ \sum_{b \in B} w(b) : B \mbox{ is a base} \} $](/files/tex/15b6767566360480b09b1982f902c978265de9c9.png)
![$ H = {\mathit stab}(S) $](/files/tex/e338cf1ff9295e47c3f1552771eb4fecfb4d730f.png)
![$$|S| \ge |H| \left( 1 - rk(M) + \sum_{Q \in G/H} rk(w^{-1}(Q)) \right).$$](/files/tex/e22b2f4b60bf6bbfd304dd219b3092fb50cbae67.png)
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Let be an (additive) abelian group, and for every
let
.