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We study the Loewy structure of the centralizer algebra kP Q for P a p-group with subgroup Q and k a field of characteristic p.Here kP Q is a special type of Hecke algebra. The main tool we employ is the decomposition kP Q = kC P (Q ) I of kP Q as a split extension of a nilpotent ideal I by the group algebra kC P (Q ).We compute the Loewy structure for several classes of groups, investigate the symmetry of the Loewy series, and give upper and lower bounds on the Loewy length of kP Q . Several of these results were discovered through the use of MAGMA, especially the general pattern for most of our computations. As a final application of the decomposition, we determine the representation type of kP Q .If G is a finite group with subgroup H and k a commutative ring with identity, then as in [7], the centralizer algebra kG H consists of all elements of kG that are invariant under the conjugation action of H . There have been several recent investigations into the representation theory of kG H in the papers [8][9][10][11]18,19]. In these papers, one of the motivating problems is the identification of the block idempotents of kG H for G a p-solvable group and H P G, or G = S n and H = S m . For P a p-group with subgroup Q and k a field of characteristic p, kP Q has no nontrivial idempotents, and therefore the questions one might ask concerning the structure and representation theory of kP Q have a somewhat different flavor than the study of the more general kG H . In particular, this paper explores the Loewy structure of kP Q and its representation type.Jennings proved in [14] a theorem that now bears his name and which allows us to the compute the radical layers of the group algebra kP for P a p-group using certain characteristic subgroups {κ i } of P . More precisely, we let κ 1 = P and inductively define κ n as the subgroup of P generated by [κ s , κ t ] whenever s, t < n and s + t n, along with all pth powers of elements from κ r whenever r < n and pr n. So κ 2 = Φ(P ) and each κ i /κ i+1 is an elementary abelian p-group. Let {x ij } s i j=1 be With the notation from Section 2, we know that J (kP Q ) = J ⊕ I where we write C = C P (Q ) and J = J (kC) for brevity. In computing J d for d > 1 it is useful to consider two separate questions: when is J I = I J ? and when is I 2 ⊆ J I ? Proposition 3.1. Let P be a p-group with subgroup Q and write k P Q = kC I. Then J I = I J if and only if Q satisfies the following condition: for all x ∈ P and all c ∈ C , there exists q ∈ Q such that [x, qc] ∈ C .Proof. To establish I J ⊆ J I it is enough to check that κ x (c −1 − 1) ∈ J I for κ x ∈ Ω and c ∈ C . We compute κ x c −1 − 1 = c −1 − 1 κ cxc −1 + (κ cxc −1 − κ x ) Since (c −1 − 1)κ cxc −1 ∈ J I we need κ cxc −1 − κ x ∈ J I . So let Ω = e i=1 Ω j be an orbit decomposition of Ω under the left action of C , so that I = kΩ j as kC -modules. Then J I =

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The subject of this paper is the investigation, load testing, and repair of a 1920's vintage public school building located in Chicago, Illinois. This school is one of thirty-two similar schools that were the subject of substantial controversy at the time they were built. Shortly after construction was completed, selected portions of the reinforced concrete roof structure were shored with supplemental steel beams to address alleged design and construction defects. The roofs have since been plagued with recurring maintenance issues related to the supplemental steel beams.An engineering study of the existing roof framing and supplemental steel reinforcement was performed to determine if the existing reinforcement was necessary. Full scale load tests of various portions of the roof structure were performed to verify the capacity of the original concrete framing. Repairs were developed to address locations where deterioration was observed or additional capacity was required, which allowed the complete removal of the existing structural steel reinforcement in place. Fiber-reinforced polymer (FRP) reinforcement, used for the first time in the Chicago Public School System, was chosen to reinforce selected concrete beams and joists. This paper will review the investigation, structural analysis, load testing and the design and installation of the externally bonded FRP reinforcement. DESCRIPTION OF STRUCTUREThe school building is a two-story reinforced concrete structure with a basement. The floor and roof construction typically consists of one-way cast-in-place reinforced concrete joists. The concrete joists are primarily supported by reinforced concrete girders and, at some locations, by the exterior masonry walls. The girders are 1991 Structures Congress 2012

Motivated by recent problems regarding the symmetry of Hecke algebras, we investigate the symmetry of the endomorphism algebra E P (M) for P a p-group and M a kP -module with k a field of characteristic p. We provide a complete analysis for cyclic p-groups and the dihedral 2-groups. For the dihedral 2-groups, this requires the classification of the indecomposable modules in terms of string modules and band modules. We generalize our techniques to consider E Λ (M) for Λ a Nakayama algebra, a local algebra, or even an arbitrary algebra.