This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$ is finite. In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material.

京都大学 / 0048 / 新制・課程博士 / 博士(医学) / 甲第20224号 / 医博第4183号 / 新制||医||1019(附属図書館) / 京都大学大学院医学研究科医学専攻 / (主査)教授 増永 慎一郎, 教授 武藤 学, 教授 羽賀 博典 / 学位規則第4条第1項該当 / Doctor of Medical Science / Kyoto University / DFAM

PURPOSE: This study was undertaken to compare HALO™ software image analysis measurements of cellularity with visual estimations from the pathologist and to outline a protocol for future experimental determinations of cellularity using HALO™. Secondly, this study investigated the clinically challenging prostate cancers of Gleason score 7 by analyzing a large database of radical prostatectomy (RP) specimens with regard to their Gleason grade composition and percentage tumor volume composition. The importance of these values of tumor cellularity, prostate volume, and tumor volume data were discussed in terms of future diagnostic endeavors. Finally, this study provided a brief background on prostate cancer, prostate cancer epidemiology, digital pathology, and the limitations and difficulties in the technological transition to digital pathology. All work for this study was done at Dana-Farber Cancer Institute (Boston, MA). METHODS: In the first part of this study, histological slides were acquired by radical prostatectomy (RP) and contained 12 tumor foci of varying degrees and sizes. These slides were scanned and imported into the HALO™ image analysis software. The tumor foci, previously demarcated by a pathologist, were annotated by hand in HALO™. An algorithm for image analysis was created by training classifiers to recognize and differentiate between epithelial tissue, stromal tissue, glass, and other. This process was accomplished by classifying 62 regions which were tested for accuracy before becoming the components of an algorithm to analyze the entire annotation layer. Each tumor focus was analyzed individually, and the results were exported into Microsoft® Excel from which relevant data were extracted. Cellularity was calculated by the percentage of tumor area that the algorithm characterized as epithelial. Cellularity values derived from HALO™ measurements for each tumor focus were compared with the visual estimations of cellularity provided by the pathologist using Pearson's correlation analysis. In the second part of this study, a database of 1386 slides containing tumors with Gleason scores between 6 and 9 was compiled from 140 RP cases. The average percentages of Gleason grades 3, 4, and 5 in each case were determined. The percentage of each slide that was occupied by the tumor was also averaged for each case, yielding an average percentage of tumor volume for each case. The average Gleason grade 3, 4, or 5 percentage for each case was plotted against the associated average tumor volume percentage of that case. The cases of Gleason score 7 (3+4, 4+3) were then isolated and plotted in a similar manner. Pearson’s correlation analysis was used to determine the degree of linear correlation between the two variables in each plot. Results: In the first part of this study, a statistically significant positive correlation between the cellularity estimations of the pathologist and the HALO™ cellularity measurements was found (r = 0.92, p < 0.01, n =12). In the second part of this study, there was a statistically significant negative correlation between average Gleason grade 3 percentage per case and average tumor volume percentage per case (r = -0.55, p <0.001, n = 140). There was also a statistically significant positive correlation between average Gleason grade 4 percentage per case and average tumor volume percentage per case (r = 0.55, p <0.001, n = 140). After slides containing Gleason score 6 (3+3) tumor were removed from the data, a statistically significant negative correlation remained between average Gleason grade 3 percentage per case and average tumor volume percentage per case (r = -0.51, p <0.001, n = 78), and a statistically significant positive correlation remained between average Gleason grade 4 percentage per case and average tumor volume percentage per case (r = 0.5, p <0.001, n = 101). A statistically significant relationship between average Gleason grade 5 percentage and average tumor volume percentage was not found (r = 0.32, p = 0.14, n = 23). CONCLUSIONS: In the first part of this study, the strong positive correlation between HALO™ cellularity values and visual estimations by the pathologist suggests that image analysis may be an effective tool for determining cellularity in digital histological images. More research using larger sample sizes is recommended to further validate the correlation between algorithm-derived cellularity from HALO™ and visual estimation by the pathologist. In the second part of this study, it appears that the volume of prostate tumors of Gleason score 7 may have prognostic power, considering that an increased percentage composition of Gleason grade 4 correlated with larger tumor volumes. Because this result may have significant clinical implications, further research specifically on tumors of Gleason score 7 is suggested to verify this relationship.

This thesis establishes a framework for cellularity of algebras related to the Jones basic construction. The framework allows a uniform proof of cellularity of Brauer algebras, BMW algebras, walled Brauer algebras, partition algebras, and others. In this setting, the cellular bases are labeled by paths on certain branching diagrams rather than by tangles. Moreover, for this class of algebras, the cellular structures are compatible with restriction and induction of modules.

The primary result from this dissertation is following inequality: d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}) in ZFC, where B is a homogeneous complete Boolean algebra, d(B) is the density, wd(B) is the weak density, and c(B) is the cellularity of B. Chapter II of this dissertation is a general overview of homogeneous complete Boolean algebras. Assuming the existence of a weakly inaccessible cardinal, we give an example of a homogeneous complete Boolean algebra which does not attain its cellularity. In chapter III, we prove that for any integer n > 1, wd_2(B) = wd_n(B). Also in this chapter, we show that if X⊂B is κ—weakly dense for 1 < κ < sat(B), then sup{wd_κ(B):κ < sat(B)} = d(B). In chapter IV, we address the following question: If X is weakly dense in a homogeneous complete Boolean algebra B, does there necessarily exist b € B\{0} such that {x∗b: x ∈ X} is dense in B|b = {c € B: c ≤ b}? We show that the answer is no for collapsing algebras. In chapter V, we give new proofs to some well known results concerning supporting antichains. A direct consequence of these results is the relation c(B) < wd(B), i.e., the weak density of a homogeneous complete Boolean algebra B is at least as big as the cellularity. Also in this chapter, we introduce discernible sets. We prove that a discernible set of cardinality no greater than c(B) cannot be weakly dense. In chapter VI, we prove the main result of this dissertation, i.e., d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}). In chapter VII, we list some unsolved problems concerning this dissertation.

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