作为 Sergeev 代数的 q-形变，Hecke-Clifford 代数是由 G. Olshanski 定义的一族有限维超代数。当 q 为 generic 时，其表示范畴半单，不可约超模对应 Schur Q-函数。本项目考虑 q=0 的退化情形，表示范畴不再半单，且同样具有丰富的组合意义。0-Hecke-Clifford 代数的有限生成模范畴的 Grothendieck 环同构于 J. Stembridge 定义的 peak 拟对称函数代数。对偶需确定其投射模范畴的 Grothendieck 环与对称群的 peak 代数之间的对应。这有助于考虑我们近期工作中提出的 Schur Q-函数拟对称加细的正性猜想及其表示论含义。另外，我们构造上述两个 Grothendieck 环之间的 Heisenberg 耦合，作为 Heisenberg 代数的推广。

As q-deformations of Sergeev algebras, Hecke-Clifford algebras are a tower of finite-dimensional superalgebras defined by G. Olshanski. When q is generic, their representation categories are semisimple with the irreducible supermodules corresponding to Schur’s Q-functions. This program considers the degenerate case when q=0, where the representation categories are no longer semisimple and also with rich combinatorial meaning. The Grothendieck ring of the category of finitely generated modules of 0-Hecke-Clifford algebras is isomorphic to the algebra of peak quasisymmetric functions defined by J. Stembridge. Dually it needs to confirm the correspondence between the Grothendieck ring of the category of projective modules and the peak algebra of symmetric groups. This helps us to consider the positivity conjecture about quasisymmetric refinements of Schur’s Q-functions proposed in our recent work for representation theoretical meaning. Also, we construct the Heisenberg double of the above two Grothendieck rings, as a generalization of the Heisenberg algebra.