The theory surrounding representations of the symmetric group is an interesting topic to study since it can be approached from several different areas of mathematics. The symmetric group of degree n (which we call Sn) can be viewed as the set of all permutations of the ordered sequence of numbers { 1, 2, …, n}. For example, with this interpretation the symmetric group of order three is the set:(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). Looking at the representation theory of the symmetric group means that we are looking at how we might translate the action of elements of this group into the language of linear algebra. It turns out that there are a finite number of irreducible “translations” (which we call representations) and that each of these can be in turn associated with a set of numbers called characters. Most of the above is traditionally studied using an area of mathematics called abstract algebra. However, one can also approach this subject using a newer area of math called combinatorics, which among other things studies the number of ways that we may count different arrangements of objects in a set. In fact, using a combinatorial algorithm called the Murnaghan Nakayama rule, we can easily generate the character table of Sn, for any n, without reference to abstract algebra or representation theory. In my project, I have been looking at ways of using combinatorics and the Murnaghan-Nakayama rule to prove certain properties about the row sums in the character table of Sn.

Diatoms, a type of phytoplankton, are microscopic organisms that live in the world’s oceans where sufficient sunlight and nutrients for photosynthesis is available. They require distinct nutrients to maximize their growth and photosynthetic ability, such as nitrogen (N). When available N is limited, diatom cells become stressed, changing their growth rates and gene expressions. In response to this stimuli, they may up- or down-regulate important genes. Such gene regulations can be determined using a tiling array, a type of microarray that uses short fragments as probes to cover the entire genome. This approach was used to test the gene expression of nutrient-replete and limited cultures of Thalassiosira pseudonana, the first diatom with a sequenced genome. Following these experiments, we chose Fragilariopsis cylindrus and Pseudo-nitzschia multiseries to investigate the effects of N-starvation on a diverse group of diatoms. In silico analysis conducted on F. cylindrus, P. multiseries and T. pseudonana identified numerous conserved genes among the three species with significant up- and down- regulation in T. pseudonana during N-limitation. Based on these results, we chose two genes of interest for further analysis. We designed primers to target and amplify UNKNOWN 4888 and CHR18 in both F. cylindrus and P. multiseries under limiting and control conditions. Changes in gene transcript accumulation were measured using quantitative PCR (qPCR). N-limited cultures of F. cylindrus and P. multiseries showed a decrease in growth rates. In F. cylindrus, UNKNOWN 4888 was down-regulated, which is consistent with our hypothesis and the pattern seen in T. pseudonana. CHR 18 showed no expression change, a result we did not expect and which was not seen in T. pseudonana. This result may be impacted by variability within the biological replicates. The gene regulations of P. multiseries are currently under investigation and will be compared to F. cylindrus and T. pseudonana.