Demazure atoms are core components in the description of Demazure crystals.  It is natural to ask to which Demazure atom a given vertex of a crystal belongs.  The answer to this question is provided by the right key map (respectively left key for opposite Demazure atoms). The effective computation of the key map has therefore captured interest across different areas.

Demazure modules were originally described as the space of global sections of a suitable line bundle on a Schubert variety. This description exhibits the natural correspondence between Schubert varieties and Demazure modules and henceforth their combinatorial skeletons, Demazure crystals or Demazure crystal atoms. In joint work with Gobet-Lecouvey,  Demazure pieces have been shown to describe RSK correspondence when rectangular nonnegative matrices are replaced  with certain general staircase-shaped matrices which permitted to study the LPP model on those matrices. More recently, Feigin-Khoroshkin-Makedonskyi have generalized the Cartan type \( A \) non-symmetric Cauchy identities by Lascoux, Azenhas-Emami, and Azenhas-Gobet-Lecouvey to arbitrary staircase-shaped matrices.

Originally, the relevance of keys stems from standard monomial theory. The Lascoux-Schutzenberger right and left keys computed via jeu de taquin on semistandard Young tableux were designed to encapsulate the Lakshmibai-Seshadri minimal respectively maximal defining chains in standard monomial theory. Since then the computation of keys has been generalized either in Cartan types or using a specific crystal model.

We present a new technique for computing the key map and the Schutzenberger-Lusztig involution using virtualization of crystals. Kashiwara introduced a method for embedding highest weight \( g_X \)-crystals inside highest weight \( g_Y \)-crystals, where \( g_D \) is the complex simple Lie algebra associated to the Dynkin diagram \( D \), provided the Dynkin diagram \( X \)  can be obtained from the Dynkin diagram \( Y \) via  a Dynkin diagram folding. This shows that key maps on crystals can be reduced to the simply-laced types. The results are type-independent and crystal model-independent. This is based on joint work with González-Huang-Torres.