Finite dimensional algebras and cellular systems

We introduce the notion of a cellular system in order to deal with quasi-hereditary algebras. We shall prove that a necessary and sufficient condition for an algebra to be quasi-hereditary is the existence of a full divisible cellular system. As a further application, we prove that the existence of a full local cellular system is a sufficient condition for a standardly stratified algebra.Comment: 15 page

Frobenius morphisms and representations of algebras

By introducing Frobenius morphisms $F$ on algebras $A$ and their modules over the algebraic closure {{\bar \BF}}_q of the finite field \BF_q of $q$ elements, we establish a relation between the representation theory of $A$ over {{\bar \BF}}_q and that of the $F$-fixed point algebra $A^F$ over \BF_q. More precisely, we prove that the category \modh A^F of finite dimensional $A^F$-modules is equivalent to the subcategory of finite dimensional $F$-stable $A$-modules, and, when $A$ is finite dimensional, we establish a bijection between the isoclasses of indecomposable $A^F$-modules and the $F$-orbits of the isoclasses of indecomposable $A$-modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over \BF_q can be interpreted as $F$-stable representations of a corresponding quiver over {{\bar \BF}}_q. We further prove that every finite dimensional hereditary algebra over \BF_q is Morita equivalent to some $A^F$, where $A$ is the path algebra of a quiver $Q$ over {{\bar \BF}}_q and $F$ is induced from a certain automorphism of $Q$. A close relation between the Auslander-Reiten theories for $A$ and $A^F$ is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of $A^F$ is obtained by "folding" the Auslander-Reiten quiver of $A$. Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver with a given dimension vector and to establish part of Kac's theorem for all finite dimensional hereditary algebras over a finite field.Comment: 28 page

Small Representations for Affine q-Schur Algebras

When the parameter $q\in\mathbb C^*$ is not a root of unity, simple modules of affine $q$-Schur algebras have been classified in terms of Frenkel--Mukhin's dominant Drinfeld polynomials (\cite[4.6.8]{DDF}). We compute these Drinfeld polynomials associated with the simple modules of an affine $q$-Schur algebra which come from the simple modules of the corresponding $q$-Schur algebra via the evaluation maps.Comment: 21 Page

Modified FEA and ExtraTree algorithm for transformer Green's function modelling

The Green's function of a transformer is essential for prediction of its vibration. As the Green's function cannot be measured directly and completely, the finite element analysis (FEA) is typically used for its estimation. However, because of the complexity of the transformer structure, the calculations involved in FEA are time consuming. Therefore, in this paper, a method based on FEA modified by an extremely random tree algorithm call ExtraTree is proposed to efficiently estimate the Green's function of a transformer. A subset of the frequency response functions from FEA will be selected by a genetic algorithm that can well present the structural variation. The FEA calculation time can be reduced by simply calculating the frequency response functions on this subset and predicting remainder using the trained ExtraTree model. The errors introduced in this method can be estimated from the corresponding frequency and the genetic algorithm error.Comment: 8 pages, 5 figures, 3 tables, submitting to internoise 201

Standard multipartitions and a combinatorial affine Schur-Weyl duality

We introduce the notion of standard multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q which is not a root of unity. We then extend the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of integral type. By the affine Schur--Weyl duality, we further extend this to a correspondence between standard multipartitions and Drinfeld multipolynomials of integral type whose associated irreducible polynomial representations completely determine all irreducible polynomial representations for the quantum loop algebra. We will see, in particular, the notion of standard multipartitions gives rise to a combinatorial description of the affine Schur--Weyl duality in terms of a column-reading vs. row reading of residues of a multipartition.Comment: 24 page

On bases of quantized enveloping algebras

We give a systematic description of many monomial bases for a given quantized enveloping algebra and of many integral monomial bases for the associated Lusztig $\mathbb Z[v,v^{-1}]$-form. The relations between monomial bases, PBW bases and canonical bases are also discussed.Comment: 13 page

Monomial bases for quantum affine sl_n

We use the idea of generic extensions to investigate the correspondence between the isomorphism classes of nilpotent representations of a cyclic quiver and the orbits in the corresponding representation varieties. We endow the set $\cal M$ of such isoclasses with a monoid structure and identify the submonoid $\cal M_c$ generated by simple modules. On the other hand, we use the partial ordering on the orbits (i.e., the Bruhat-Chevalley type ordering) to induce a poset structure on $\cal M$ and describe the poset ideals generated by an element of the submonoid $\cal M_c$ in terms of the existence of a certain composition series of the corresponding module. As applications of these results, we generalize some results of Ringel involving special words to results with no restriction on words and obtain a systematic description of many monomial bases for any given quantum affine ${\frak {sl}}_n$.Comment: 24 page

Ariki-Koike Algebras with Semisimple Bottoms

We investigated the representation thoery of an Ariki-Koike algebra whose Poincare polynomial associated with the "bottom", i.e., the subgroup on which the symmetric group acts, is non-zero in the base field. We proved that the module category of such an Ariki-Koike algebra is Morita equivalent to the module category of a direct sum of tensor products of Hecke algebras associated with certain symmetric groups. We also generalized this Morita equivalence theorem to give a Morita equivalenve between a $q$-Schur$^m$ algebra and a direct sum of tensor products of certain $q$-Schur algebras.Comment: 20 pages. Math. Zeit. (to appear

Canonical bases for the quantum linear supergroups

We give a combinatorial construction for the canonical bases of the $\pm$-parts of the quantum enveloping superalgebra \bfU(\mathfrak{gl}_{m|n}) and discuss their relationship with the Kazhdan-Lusztig bases for the quantum Schur superalgebras \bsS(m|n,r) introduced in \cite{DR}. We will also extend this relationship to the induced bases for simple polynomial representations of \bfU(\mathfrak{gl}_{m|n}).Comment: 30 page

The queer q-Schur superalgebra

As a natural generalization quantum Schur algebras associated with the Hecke algebra of the symmetric group, we introduce the quantum Schur superalgebra of type Q associated with the Hecke-Clifford superalgebra, which, by definition, is the endomorphism algebra of the induced module over the Hecke-Clifford superalgebra from certain permutation modules over the Hecke algebra of the symmetric group. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base change property for them. We will also identify the quantum Schur superalgebra of type Q with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and then provide a classification of their irreducible representations over a certain extension of the field of complex rational functions.Comment: 27 pages, version 2 with an appendix added, to appear in Journal of the Australian Mathematical Society, available at https://doi.org/10.1017/S144678871700033