Logarithmic concave sequences are ubiquitous and special in nature (for example, binomial coefficients, Sterling numbers, graph matching numbers, unsigned coefficients of the reduced chromatic polynomials of matriods etc.) It is then interesting to ask whether a log-concave sequence behaves nicely with respect to the symmetry of the underlying mathematical object. The notion of equivariant log-concavity was introduced by Gedeon, Proudfoot and Young in the context of matroids. We will highlight some known results about equivariant log-concavity. As an example of an interplay between algebraic geometry and combinatorics, I will show that the graded vector space spanned by independent vertex sets of any claw-free graph is equivariantly log-concave. Our proof reduces the problem to the equivariant hard Lefschetz theorem on the cohomology of a product of projective lines. Both the result and the proof generalize our previous result on graph matchings. This also gives a strengthening and a new proof of results of Hamidoune, and Chudnovsky--Seymour.