# Algebraic combinatorics and coinvariant spaces by Francois Bergeron

By Francois Bergeron

Written for graduate scholars in arithmetic or non-specialist mathematicians who desire to study the fundamentals approximately the most vital present examine within the box, this publication presents a thorough, but available, creation to the topic of algebraic combinatorics. After recalling uncomplicated notions of combinatorics, illustration concept, and a few commutative algebra, the most fabric presents hyperlinks among the research of coinvariant or diagonally coinvariant areas and the examine of Macdonald polynomials and similar operators. this provides upward push to a number of combinatorial questions in relation to items counted by means of standard numbers resembling the factorials, Catalan numbers, and the variety of Cayley bushes or parking services. the writer bargains rules for extending the idea to different households of finite Coxeter teams, along with permutation teams.

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1998 . .. a good index is incorporated to be able to aid a mathematician operating in a space except his personal to discover adequate details at the subject in query.

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In the even more special case when a1 a2 · · · an is a permutation of {1, 2, . . , n}, the tableau P is also standard. This establishes a bijection between permutations in SN and pairs of standard tableaux of the same shape. As a biproduct, we see that (f μ )2 , n! 6) μ n since (f μ )2 counts the number of pairs of standard tableaux of shape μ. The inverse u−1 of a lexicographic word of biletters u= b1 a1 ··· ··· b2 a2 bk ak is the biletter lexicographic word u−1 = lex a1 b1 a2 b2 ··· ··· ak , bk with “lex(v)” standing for the increasing lexicographic reordering of v.

1 Tableaux We now come to another central notion, that of tableaux. Among several other roles they play, a crucial one is the construction of irreducible representations of the symmetric group. The construction in question involves the calculation of certain polynomials associated with “ﬁllings” of the cells of a diagram. Such ﬁllings are called tableaux. More precisely, a tableau τ of shape d with values in a set A (usually some subset of N) is a function τ : d −→ A. We denote by λ(τ ) the shape d of τ , and we think of τ (c) as being the entry or value that appears in cell c.

For more on this see chapter 6 of [Lothaire 02]. “berg” — 2009/4/13 — 13:55 — page 45 — #53 Chapter 3 Invariant Theory It is now time to start mixing algebraic components into our algebraicocombinatorial recipe. The ﬁrst of these ingredients comes from invariant theory with some special emphasis on symmetric polynomials. Much of this theory makes systematic use of combinatorial objects, both in the formulation of the main results and in the proof techniques. To learn more about the notions discussed here, refer to [Humphreys 90] and the “bible” of symmetric functions [Macdonald 95].