Titles and Short Summaries of the Talks

18 Algebra
Final: 2013/2/7
35 Mitsuhiro Miyazaki
(Kyoto Univ. of Edu.)
∗ Tensor of indeterminates and invariant theory · · · · · · · · · · · · · · · · · · · · · 10
Summary: High-dimensional array data analysis is now rapidly developing and being successfully
applied in various fields. A high-dimensional array datum is called a tensor in those communities.
To be precise, a d-dimensional array datum (ai1i2···id )1≤ij≤mj is called a d-tensor or an m1×· · ·×md-
tensor.
A 2-tensor is no other than a matrix. For a matrix of indeterminates, that is, a matrix whose entries
are independent indeterminates, there are many results about the action of classical groups and the
ring of invariants under these actions. In this talk, we consider an m×n×2-tensor of indeterminates
and the action of the product of special linear groups on the polynomial ring generated by the
entries of the tensor.
36 Kyouko Kimura (Shizuoka Univ.) ∗ Non-vanishingness of Betti numbers of edge ideals and complete bipartite
graphs · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 10
Summary: We study the non-vanishingness of Betti numbers of edge ideals. Let G be a finite
simple graph. Then we can associate an ideal I(G) with G, which is called the edge ideal of G.
We show that the Betti number of I(G) does not vanish if G contains a set of complete bipartite
subgraphs with some properties, one of which is related to the 3-disjointness of edges of G.
37 Takao Hayami (Hokkai-Gakuen Univ.) ∗ Hochschild cohomology ring of quaternion algebras · · · · · · · · · · · · · · · · · 10
Summary: We give an efficient bimodule projective resolution of the generalized quaternion Z
algebra Γ. As a main result, we determine the ring structure of the Hochschild cohomology HH ∗ (Γ)
by calculating the Yoneda products using this resolution.
38 Kenichi Shimizu (Nagoya Univ.) ♯ On indicators of Hopf algebras · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 15
Summary: Kashina, Montgomery and Ng introduced the n-th indicator νn(H) of a finite-
dimensional Hopf algebra H and showed that the indicators have some interesting properties
such as the gauge invariance. I will talk about recent progress on the study of the properties of the
indicators, including (1) the cyclotomic integrality, (2) a formula for the opposite Hopf algebra, (3)
a formula for the Drinfeld double, and (4) a relation to the quasi-exponent.
39 Hiroki Sasaki (Shinshu Univ.) ♯ Cohomology rings of tame blocks · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 15
Summary: We give a theorem concerning direct summands of source algebras of block ideals.
We also show that the cohomology rings of tame blocks are the images of transfer maps from the
cohomology rings of their defect groups induced by direct summands of source algebras.
40 Tsuyoshi Miezaki (Yamagata Univ.) ♯
Thomas Creutzig (TU Darmstadt)
Gerald Höhn (Kansas State Univ.)
The McKay–Thompson series of Mathieu Moonshine modulo two · · · · 10
Summary: In this note, we describe the parity of the coefficients of the McKay–Thompson series
of Mathieu moonshine. As an application, we prove a conjecture of Cheng, Duncan and Harvey
stated in connection with umbral moonshine for the case of Mathieu moonshine.