Katsuhiro Uno

On representations of non-semisimple specialized Hecke algebras

where I is the length function. See [4, Chap. 4, Sect. 2, Ex. 23; 151. Let cr be a complex number. Then we obtain a C-algebra H,(W) by specialization q H a. (See p. 637 of [7, II . Note: We actually consider the specialization &H & choosing 2 c(. However, we indicate it by q++ c1 for convenience.) Suppose that ( W, S) is not of type A, x . x A 1. Then it is known that H, ( W) is semisimple if and only if CI # 0 and Pw(a) # 0, where P,(q) is the Poincart polynomial of (W, S) defined by P&q)= x3,., w qtcw). (See [ 13, Theorem].) Moreover, if H,( W) is semisimple, then it is isomorphic to the group algebra C(W). (See [7, Sect. 681.) Then one might be interested in what representations of non-semisimple specialized Hecke algebras look like. After we work with some examples, we recognize that in certain cases their representations quite resemble those of W over some finite field: Their indecomposable modules, the Loewy structure, the block decomposition, and so on. In particular, if ( W, S) is of type A,_ 1 and CI is a primitive pth root of unity for some prime p, then, at least for small p, we find by direct computation that representations of H,(W) look

Character relations and simple modules in the Auslander-Reiten graph of the symmetric and alternating groups and their covering groups

From character relations for symmetric groups or Hecke algebras such as the Murnaghan–Nakayama formula and the Jantzen–Schaper formula, we obtain a lower bound for the diagonal entries of Cartan matrices. Moreover, we prove an analogous character relation for covering groups of symmetric groups and obtain a similar lower bound. As an application, we show in these situations that for wild blocks simple modules must lie at the end of the Auslander–Reiten quiver, which is equivalent to the fact that the hearts of projective indecomposable modules are indecomposable.

An Investigation of Prospective Elementary Teachers’ Argumentation from the Perspective of Mathematical Knowledge for Teaching and Evaluating

The purpose of this study is to investigate prospective elementary school teachers’ mathematical process knowledge related to argumentation. To achieve this, we focus on prospective teachers’ mathematical argumentation as a key aspect of the mathematical knowledge teachers need for teaching. By referring to the framework of mathematical knowledge for teaching, we pay special attention to “process knowledge” instead of “content knowledge.” The study involves 136 prospective teachers at a national university in Japan who performed a task requiring the evaluation of several incorrect solutions to a realistic problem. The results show that most prospective teachers have difficulties in evaluating or assessing children’s incorrect solutions. This study contributes to the field on a conceptual and a methodological level. Regarding the conceptual framework, we suggest the importance of teachers’ process knowledge in teaching and evaluating, particularly in relation to mathematical argumentation and, regarding methodology, we create a way to help participants notice children’s incomplete thinking.

Auslander-Reiten sequences for certain group modules

Let kG be the group algebra of a finite group G over a field k of characteristic p, where p is a prime. Fix a non-trivial normal p-subgroup Q of G. In this paper, we study Auslander-Reiten sequences terminating at indecomposable kc-modules whose vertices are Q and whose sources are factor modules of kQ by some power of its radical. In particular, we consider relative projectivity of the middle terms of those sequences. In our main theorem, we shall prove that, for any subgroup H of G with Q < H < G, the middle term is H-projective if and only if the head of the third term is. We can also show that the head of the third term and some indecomposable direct summand of the middle term have vertices in common unless the former is Q-projective. In the case where Q is cyclic, these can be applied to all the AuslanderReiten sequences terminating at indecomposable kG-modules with vertices Q, since any such module has a source of the above form. The result in this case is related to a recent work of Erdmann [3] (see Section 3). The proof of the if part of the theorem is a direct consequence of [S, Theorem 5.41, whereas the only if part, whose proof is the main body of the present paper, can be proved by easy calculations. Notation is standard. For an indecomposable kG-module W, a vertex of W is denoted by ux( W). If a kG-module W’ is isomorphic to a direct summand of another kG-module IV”, we write IV’1 W”. Also Horn&W’, W”) is simply denoted by (IV’, W”) for convenience. J and J,, mean the Jacobson radicals of kG and kQ, respectively. Note that J,, is just the augmentation ideal of kQ. For any non-projective indecomposable kG-module W,