Supercharacters and the discrete Fourier, cosine, and sine transforms

References

• Aguiar, M., André, C., Benedetti, C., Bergeron, N., Chen, Z., Diaconis, P., Hendrickson, A., Hsiao, S., Isaacs, I. M., Jedwab, A., Johnson, K., Karaali, G., Lauve, A., Le, T., Lewis, S., Li, H., Magaard, K., Marberg, E., Novelli, J.-C., Pang, A., Saliola, F., Tevlin, L., Thibon, J.-Y., Thiem, N., Venkateswaran, V., Vinroot, C. R., Yan, N., Zabrocki, M. (2012). Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras. Adv. Math. 229(4):2310–2337. MR 2880223
   Web of Science ®Google Scholar
• André, C. A. M. (1995). Basic characters of the unitriangular group. J. Algebra 175(1):287–319. MR 1338979 (96h:20081a)
   Web of Science ®Google Scholar
• André, C. A. M. (2001). The basic character table of the unitriangular group. J. Algebra 241(1):437–471. MR 1839342 (2002e:20082)
   Web of Science ®Google Scholar
• André, C. A. M. (2002). Basic characters of the unitriangular group (for arbitrary primes). Proc. Am. Math. Soc. 130(7):1943–1954 (electronic). MR 1896026 (2003g:20075)
   Web of Science ®Google Scholar
• Benidt, S. G., Hall, W. R. S., Hendrickson, A. O. F. (2014). Upper and lower semimodularity of the supercharacter theory lattices of cyclic groups. Comm. Algebra 42(3):1123–1135. MR 3169622
   Web of Science ®Google Scholar
• Berndt, B. C., Evans, R. J., Williams, K. S. (1998). Gauss and Jacobi Sums. Canadian Mathematical Society Series of Monographs and Advanced Texts. New York: John Wiley & Sons, Inc. (A Wiley-Interscience Publication). MR 1625181.
   Google Scholar
• Bini, D., Capovani, M. (1983). Spectral and computational properties of band symmetric Toeplitz matrices. Spectral and computational properties of band symmetric Toeplitz matrices 52/53:99–126. MR 709346
   Google Scholar
• Boman, E., Koltracht, I. (1995). Fast transform based preconditioners for Toeplitz equations. SIAM J. Matrix Anal. Appl. 16(2):628–645. MR 1321801
   Web of Science ®Google Scholar
• Brumbaugh, J. L., Bulkow, M., Fleming, P. S., Alberto, L., German, G., Garcia, S. R., Karaali, G., Michal, M., Turner, A. P., Suh, H. (2014). Supercharacters, exponential sums, and the uncertainty principle. J. Number Theory 144:151–175. MR 3239156
   Web of Science ®Google Scholar
• Brumbaugh, J. L., Bulkow, M., German, L. A. G., Garcia, S. R., Michal, M., Turner, A. P. (2013). The graphic nature of the symmetric group. Exp. Math. 22(4):421–442. MR 3171103
   Web of Science ®Google Scholar
• Chan, R. H., Ng, M. K., Wong, C. K. (1996). Sine transform based preconditioners for symmetric Toeplitz systems. Linear Algebra Appl. 232:237–259. MR 1366587
   Web of Science ®Google Scholar
• Cooley, J. W., Tukey, J. W. (1965). An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19:297–301. MR 0178586
   Web of Science ®Google Scholar
• Di Fiore, C., Zellini, P. (1995). Matrix decompositions using displacement rank and classes of commutative matrix algebras. Linear Algebra Appl. 229:49–99. MR 1352839
   Web of Science ®Google Scholar
• Diaconis, P., Isaacs, I. M. (2008). Supercharacters and superclasses for algebra groups. Trans. Am. Math. Soc. 360(5):2359–2392. MR 2373317
   Web of Science ®Google Scholar
• Diaconis, P., Thiem, N. (2009). Supercharacter formulas for pattern groups. Trans. Am. Math. Soc. 361(7):3501–3533. MR2491890 (2010g:20013)
   Web of Science ®Google Scholar
• Duke, W., Garcia, S. R., Lutz, B. (2015). The graphic nature of Gaussian periods. Proc. Am. Math. Soc. 143(5):1849–1863. MR 3314096
   Web of Science ®Google Scholar
• Feig, E., Ben-Or, M. (1997). On algebras related to the discrete cosine transform. Linear Algebra Appl. 266:81–106. MR 1473195
   Web of Science ®Google Scholar
• Fowler, C. F., Garcia, S. R., Karaali, G. (2014). Ramanujan sums as supercharacters. Ramanujan J. 35(2):205–241. MR 3266478
   Web of Science ®Google Scholar
• Garcia, S. R., Horn, R. A. (2017). A Second Course in Linear Algebra. Cambridge Mathematical Textbooks. New York, NY: Cambridge University Press.
   Google Scholar
• Garcia, S. R., Hyde, T., Lutz, B. (2015). Gauss’s hidden menagerie: From cyclotomy to supercharacters. Notices Am. Math. Soc. 62(8):878–888. MR 3379072
   Google Scholar
• Garcia, S. R., Lutz, B. A supercharacter approach to Heilbronn sums. J. Number Theory (in press).
   Google Scholar
• Gonzalez, R. C., Woods, R. E. (2017). Digital Image Processing, 4th ed. New York: Pearson.
   Google Scholar
• Grishin, D., Strohmer, T. (2004). Fast scattered data approximation with Neumann and other boundary conditions. Linear Algebra Appl. 391:99–123. MR 2094605
   Web of Science ®Google Scholar
• Heideman, M. T., Johnson, D. H., Burrus, C. S. (1985). Gauss and the history of the fast Fourier transform. Arch. Hist. Exact Sci. 34(3):265–277. MR 815154
   Web of Science ®Google Scholar
• Heinig, G. (2001). Chebyshev-Hankel matrices and the splitting approach for centrosymmetric Toeplitz-plus-Hankel matrices. Linear Algebra Appl. 327(1-3):181–196. MR 1823348
   Google Scholar
• Hendrickson, A. O. F. (2008). Supercharacter theories of cyclic p-groups. Ph.D. Thesis. ProQuest LLC, Ann Arbor, MI, The University of Wisconsin-Madison. MR 2711764
   Google Scholar
• Hendrickson, A. O. F. (2012). Supercharacter theory constructions corresponding to Schur ring products. Comm. Algebra 40(12):4420–4438. MR 2989654
   Web of Science ®Google Scholar
• Hendrickx, J., Van Barel, M. (2002). Fast direct solution methods for symmetric banded Toeplitz systems, based on the sine transform. Fast direct solution methods for symmetric banded Toeplitz systems, based on the sine transform 343/344:211–232 (Special issue on structured and infinite systems of linear equations). MR 1878943
   Google Scholar
• Humphries, S. P., Johnson, K. W. (2008). Fusions of character tables and Schur rings of abelian groups. Comm. Algebra 36(4):1437–1460. MR 2406596 (2009b:20008)
   Web of Science ®Google Scholar
• Kammler, D. W. (2007). A First Course in Fourier Analysis, 2nd ed. Cambridge: Cambridge University Press. MR 2382058
   Google Scholar
• Sanchez, V., Garcia, P., Peinado, A., Segura, J., Rubio, A. (1995). Diagonalizing properties of the discrete cosine transforms. IEEE Trans. Signal Process. 43(11):2631–2641.
   Web of Science ®Google Scholar
• Sanchez, V., Peinado, A. M., Segura, J. C., Garcia, P., Rubio, A. J. (1996). Generating matrices for the discrete sine transforms. IEEE Trans. Signal Process. 44(10):2644–2646.
   Web of Science ®Google Scholar
• Stein, E. M., Shakarchi, R. (2003). Fourier Analysis. Princeton Lectures in Analysis, Vol. 1. Princeton, NJ: Princeton University Press (an introduction). MR 1970295
   Google Scholar
• Sullivan, F., Dongarra, J. (2000). Guest editors’ introduction: The top 10 algorithms. Comput. Sci. Eng. 2:22–23.
   Web of Science ®Google Scholar
• Thiem, N. (2010). Branching rules in the ring of superclass functions of unipotent upper-triangular matrices. J. Algebraic Combin. 31(2):267–298. MR 2592079
   Web of Science ®Google Scholar
• Thiem, N., Venkateswaran, V. (2009). Restricting supercharacters of the finite group of unipotent uppertriangular matrices. Electron. J. Combin. 16(1):32 (Research Paper 23). MR 2482091 (2010e:20024)
   Google Scholar