Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Green, Richard Mutegeki
Languages: English
Types: Doctoral thesis
Subjects: QA

Classified by OpenAIRE into

arxiv: Mathematics::Representation Theory
The main aim of this thesis is to investigate the relationship between the quantized enveloping algebra U(gln) (corresponding to the Lie algebra gln) and the q-Schur algebra, Sq(n, r). It was shown in [BLM] that there is a surjective algebra homomorphism\ud \ud θr : (gln)→Z[v, v -1] ⓍSq(n,r),\ud \ud where q = v2. \ud §1 is devoted to background material.\ud In §2, we show explicitly how to embed the q-Schur algebra into the r-th tensor power of a suitable n x n matrix ring. This gives a product rule for the q-Schur algebra with similar properties to Schur's product rule for the unquantized Schur algebra. A corollary of this is that we can describe, in §2.3, a certain family of subalgebras of the q-Schur algebra.\ud \ud In §3, we use the product rule of §2 to prove a q-analogue of Woodcock's straightening formula for codeterminants. This gives a basis of "standard quantized codeterminants" for Sq(n, r) which is heavily used in chapters 4, 5 and 6.\ud \ud In §4, we use the theory of quantized codeterminants developed in §3 to describe preimages under the homomorphism Or and the kernel of Or.\ud \ud In §5, we use the results of §3 and §4 to link the representation theories of U(gln} and Sq(n, r). We also obtain a simplified proof of Dipper and James' "semistandard basis theorem" for q- Weyl modules of q-Schur algebras.\ud \ud In §6, we show how to make the set of q-Schur algebras Sq(n, r) (for a fixed n) into an inverse system. We prove that the resulting inverse limit, Sv(n), is a cellular algebra which is closely related to the quantized enveloping algebra U(sln) and Lusztig's algebra U.\ud

Share - Bookmark

Cite this article