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23 pagesInternational audienceWe identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras

Abstract. Let Un(q) denote the (upper) unitriangular group of degree n over the finite field Fq with q elements. In this paper we consider the basic (complex) characters of Un(q) and we prove that every irreducible (complex) character of Un(q) is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption p ≥ n, where p is the characteristic of the field Fq.

We define and study supercharacters of the classical finite unipotent groups of types B n (q), C n (q) and D n (q). We show that the results we proved in 2006 remain valid over any finite field of odd characteristic. In particular, we show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. In addition, we prove that the unitary vector space spanned by all the supercharacters is closed under multiplication, and establish a formula for the supercharacter values. As a consequence, we obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we give a combinatorial description of all the irreducible characters of maximum degree in terms of the root system, by showing how they can be obtained as constituents of particular supercharacters.

We define the superclasses for a classical finite unipotent group U of type B n (q), C n (q), or D n (q), and show that, together with the supercharacters defined in [C.A.M. André, A.M. Neto, Supercharacters of the Sylow p-subgroups of the finite symplectic and orthogonal groups, Pacific J. Math. 239 (2) (2009) 201-230], they form a supercharacter theory in the sense of [P. Diaconis, I.M. Isaacs, Supercharacters and superclasses for algebra groups, Trans. Amer. Math. Soc. 360 (5) (2008) 2359-2392].In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As a consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of U , and prove various orthogonality rela-✩ This research was made within the activities

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