京都大学数理解析研究所

Title

(talk 1) Geometric Complexity Theory
(talk 2) The quantum deformation of the restriction of GL_{mn}-modules to GL_m \times GL_n

Abstract

　 (talk 1)
Let X be an m-by-m matrix and consider the (i) the form det(X), the determinant,and, (ii) perm(X), the permanent (i.e., the determinant without the signs).
There are efficient algorithms to compute the determinant, while no such algorithm is known for the permanent. P v. NP and other questions in theoretical computer science ask for such proofs.

In our approach, we convert the non-existence of an algorithm to the existence of "obstructions" or "witnesses", which are constructs from geometric invariant theory and ultimately representation theory. We show that the GL(X) orbits, and their closures, of det(X), perm(X) have a lot to do with the question. We show that both det(X) and perm(X) are GIT-stable. Since both the forms have a very distinctive stabilizers H in GL(X), this immediately brings us to the Peter-Weyl modules, i.e. GL(X)-modules with an H-invariant vector. The witnesses or obstructions are from such modules.

We also survey another approach to the problem and analyse its GIT content. Finally, we pose the group restriction problem, i.e., the presence of H-invariant vectors in a G-module, as the main problem. We outline why quantum algebras should be the key tool for the problem.

This is joint work with Ketan Mulmuley.


(talk 2)
We consider the problem of classification of P-W modules for a pair (H,G), as posed earlier. We quickly survey the key results in the (G, G \times G), which is now well-understood in its algebraic and combinatorial facets. We hope for a similar insight for the stabilizer of the determinant, i.e., (GL_m \times GL_m , GL_{m^2 }), where the embedding is
(A,B)(X) \rightarrow AXB^{-1}.

We consider the (slightly more general) embedding
GL_m \times GL_n \rightarrow GL_{mn}.

A quantization of the above embedding is not known. However, for any Weyl module $V_{\lambda }(\C^mn)$ of $GL_{mn}$ we construct a $U_q (gl_m ) \otimes U_q (gl_n )$-structure, which at q=1 matches the classical embedding.

This is done in two steps: (i) A faithful embedding of $U_q (gl_m ) \otimes U_q (gl_n ) \rightarrow U_q (gl_{mn})$ for the action on $\wedge^k (\C^mn)$. (ii) a straightening law which helps construct the general $V_{\lambda }(\C^mn)$. For the $\wedge^k (\C^mn)$ we also exhibit the bi-crystal structure, thereby constructing the algebra behind the combinatorial results of Danilov et al.

The general problem is to construct a bi-crystal structure on semistandard tableaus with entries in [mn]. Our earlier result proves that such a structure exists. We demonstrate a candidate U_q (gl_m ) crystal structure for n=2 and ask if this is what arises from the above quantization.

This is ongoing joint work with Bharat Adsul and K. V. Subrahmanyam.

Date

June 18 (Thu), 10:30-12:30

Room

RIMS room 102 (いつもと部屋が違いますのでご注意下さい)

Speaker

Prof. Hiraku Nakajima (RIMS)