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The answer to the question is "yes", allowing for a generous interpretation of "direct way". This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of coefficients of Kazhdan-Lusztig polynomials for an arbitrary Coxeter group here.

See the Update to my MO question here, which refers to the 1991 conference report by Jim Carrell (with Dale Peterson): in the first section, the equivalence you want is formulated for an arbitrary Coxeter group under the hypothesis that coefficients of relevant K-L polynomials are all non-negative. (This may be one of the sources you are referring to.)

At first I had overlooked this type of answer to my own question. (I'm still looking for other consequences of the non-negativity theorem, of course, but this one is interesting.) Note that for general Coxeter groups, one needs an approach which doesn't involve the geometry of Schubert varieties. What Elias and Williamson seem to do is avoid all that algebraic geometry by providing a sophisticated substitute.

The answer to the question is "yes", allowing for a generous interpretation of "direct way". This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of coefficients of Kazhdan-Lusztig polynomials for an arbitrary Coxeter group here.

See the Update to my MO question here, which refers to the 1991 conference report by Jim Carrell (with Dale Peterson): in the first section, the equivalence you want is formulated for an arbitrary Coxeter group under the hypothesis that coefficients of relevant K-L polynomials are all non-negative. (This may be one of the sources you are referring to.)

At first I had overlooked this type of answer to my own question. (I'm still looking for other consequences of the non-negativity theorem, of course, but this one is interesting.) Note that for general Coxeter groups, one needs an approach which doesn't involve the geometry of Schubert varieties. What Elias and Williamson seem to do is avoid all that algebraic geometry by providing a sophisticated substitute.

The answer to the question is "yes", allowing for a generous interpretation of "direct way". This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of coefficients of Kazhdan-Lusztig polynomials for an arbitrary Coxeter group here.

See the Update to my MO question here, which refers to the 1991 conference report by Jim Carrell (with Dale Peterson): in the first section, the equivalence you want is formulated for an arbitrary Coxeter group under the hypothesis that coefficients of relevant K-L polynomials are all non-negative. (This may be one of the sources you are referring to.)

At first I had overlooked this type of answer to my own question. (I'm still looking for other consequences of the non-negativity theorem, of course, but this one is interesting.) Note that for general Coxeter groups, one needs an approach which doesn't involve the geometry of Schubert varieties. What Elias and Williamson seem to do is avoid all that algebraic geometry by providing a sophisticated substitute.

The answer to the question is "yes", allowing for a generous interpretation of "direct way". This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of coefficients of Kazhdan-Lusztig polynomials for an arbitrary Coxeter group here.

See the Update to my MO question here, which refers to the 1991 conference report by Jim Carrell (with Dale Peterson): in the first section, the equivalence you want is formulated for an arbitrary Coxeter group under the hypothesis that coefficients of relevant K-L polynomials are all non-negative. (This may be one of the sources you are referring to.)

At first I had overlooked this type of answer to my own question. (I'm still looking for other consequences of the non-negativity theorem, of course, but this one is interesting.) Note that for general Coxeter groups, one needs an approach which doesn't involve the geometry of Schubert varieties. What Elias and Williamson seem to do is avoid all that algebraic geometry by providing a sophisticated substitute.

The answer to the question is "yes", allowing for a generous interpretation of "direct way". This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of coefficients of Kazhdan-Lusztig polynomials for an arbitrary Coxeter group here.

See the Update to my MO question here, which refers to the 1991 conference report by Jim Carrell (with Dale Peterson): in the first section, the equivalence you want is formulated for an arbitrary Coxeter group under the hypothesis that coefficients of relevant K-L polynomials are all non-negative. (This may be one of the sources you are referring to.)

At first I had overlooked this type of answer to my own question. Jim Carrell may want to elaborate further. (I'm still looking for other consequences of the non-negativity theorem, of course, but this one is interesting.)

The answer to the question is "yes", allowing for a generous interpretation of "direct way". This will follow from the recently posted work of Ben Elias and Geordie Williamson on non-negativity of coefficients of Kazhdan-Lusztig polynomials for an arbitrary Coxeter group here.

See the Update to my MO question here, which refers to the 1991 conference report by Jim Carrell (with Dale Peterson): in the first section, the equivalence you want is formulated for an arbitrary Coxeter group under the hypothesis that coefficients of relevant K-L polynomials are all non-negative. (This may be one of the sources you are referring to.)

At first I had overlooked this type of answer to my own question. (I'm still looking for other consequences of the non-negativity theorem, of course, but this one is interesting.) Note that for general Coxeter groups, one needs an approach which doesn't involve the geometry of Schubert varieties. What Elias and Williamson seem to do is avoid all that algebraic geometry by providing a sophisticated substitute.