6,844 research outputs found

### On the endomorphisms of Weyl modules over affine Kac-Moody algebras at the critical level

We present an independent short proof of the main result of arXiv:0706.3725
that the algebra of endomorphisms of a Weyl module of critical level is
isomorphic to the algebra of functions on the space of monodromy-free opers on
the disc with regular singularity and residue determined by the highest weight
of the Weyl module. We derive this from the results of arXiv:0712.1183 about
the shift of argument subalgebras.Comment: 9 page

### Polynomial identities for matrices over the Grassmann algebra

We determine minimal Cayley--Hamilton and Capelli identities for matrices
over a Grassmann algebra of finite rank. For minimal standard identities, we
give lower and upper bounds on the degree. These results improve on upper
bounds given by L.\ M\'arki, J.\ Meyer, J.\ Szigeti, and L.\ van Wyk in a
recent paper.Comment: 9 page

### Remarks on the $\alpha$--permanent

We recall Vere-Jones's definition of the $\alpha$--permanent and describe the
connection between the (1/2)--permanent and the hafnian. We establish expansion
formulae for the $\alpha$--permanent in terms of partitions of the index set,
and we use these to prove Lieb-type inequalities for the $\pm\alpha$--permanent
of a positive semi-definite Hermitian $n\times n$ matrix and the
$\alpha/2$--permanent of a positive semi-definite real symmetric $n\times n$
matrix if $\alpha$ is a nonnegative integer or $\alpha\ge n-1$. We are unable
to settle Shirai's nonnegativity conjecture for $\alpha$--permanents when
$\alpha\ge 1$, but we verify it up to the $5\times 5$ case, in addition to
recovering and refining some of Shirai's partial results by purely
combinatorial proofs.Comment: 9 page

### Comments on the Deformed W_N Algebra

We obtain an explicit expression for the defining relation of the deformed
W_N algebra, DWA(^sl_N)_{q,t}. Using this expression we can show that, in the
q-->1 limit, DWA(^sl_N)_{q,t} with t=e^{-2\pi i/N}q^{(k+N)/N} reduces to the
sl_N-version of the Lepowsky-Wilson's Z-algebra of level k, ZA(^sl_N)_k. In
other words DWA(^sl_N)_{q,t} with t=e^{-2\pi i/N}q^{(k+N)/N} can be considered
as a q-deformation of ZA(^sl_N)_k. In the appendix given by H.Awata, S.Odake
and J.Shiraishi, we present an interesting relation between DWA(^sl_N)_{q,t}
and \zeta-function regularization.Comment: 10 pages, LaTeX2e with ws-ijmpb.cls, Talk at the APCTP-Nankai Joint
Symposium on ``Lattice Statistics and Mathematical Physics'', 8-10 October
2001, Tianjin Chin

- …