Murnaghan-Nakayama Rule Research Articles

A Murnaghan–Nakayama rule for noncommutative Schur functions

We prove a Murnaghan–Nakayama rule for noncommutative Schur functions introduced by Bessenrodt, Luoto and van Willigenburg. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan–Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients ±1 determined by the height of these border strips. The rule is proved by interpreting the noncommutative Pieri rules for noncommutative Schur functions in terms of box-adding operators on compositions.

A Murnaghan–Nakayama rule for values of unipotent characters in classical groups

We derive a Murnaghan–Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai’s explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some ℓ \ell -singular element for certain primes ℓ \ell . As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes ℓ ≥ 3 \ell \ge 3 the first Cartan invariant in the principal ℓ \ell -block is larger than 2 unless Sylow ℓ \ell -subgroups are cyclic.

Another Proof of the Harer-Zagier Formula

For a regular $2n$-gon there are $(2n-1)!!$ ways to match and glue the $2n$ sides. The Harer-Zagier bivariate generating function enumerates the gluings by $n$ and the genus $g$ of the attendant surface and leads to a recurrence equation for the counts of gluings with parameters $n$ and $g$. This formula was originally obtained using multidimensional Gaussian integrals. Soon after, Jackson and later Zagier found alternative proofs using symmetric group characters. In this note we give a different, characters-based, proof. Its core is computing and marginally inverting the Fourier transform of the underlying probability measure on $S_{2n}$. A key ingredient is the Murnaghan-Nakayama rule for the characters associated with one-hook Young diagrams.

The role of residue and quotient tables in the theory of k-Schur functions

Recently, residue and quotient tables were defined by Fishel and the author, and were used to describe strong covers in the lattice of k-bounded partitions. In this paper, we prove (and, in some cases, conjecture) that residue and quotient tables can be used to describe many other results in the theory of k-bounded partitions and k-Schur functions, including k-conjugates, weak horizontal and vertical strips, and the Murnaghan–Nakayama rule. Evidence is presented for the claim that one of the most important open questions in the theory of k-Schur functions, a general rule that would describe their product, can be also concisely stated in terms of residue tables.

Character deflations and a generalization of the Murnaghan–Nakayama rule

Abstract Given natural numbers m and n, we define a deflation map from the characters of the symmetric group S mn to the characters of Sn . This map is obtained by first restricting a character of S mn to the wreath product Sm ≀ Sn , and then taking the sum of the irreducible constituents of the restricted character on which the base group Sm × ⋯ × Sm acts trivially. We prove a combinatorial formula which gives the values of the images of the irreducible characters of S mn under this map. We also prove an analogous result for more general deflation maps in which the base group is not required to act trivially. These results generalize the Murnaghan–Nakayama rule and special cases of the Littlewood–Richardson rule. As a corollary we obtain a new combinatorial formula for the character multiplicities that are the subject of the long-standing Foulkes' Conjecture. Using this formula we verify Foulkes' Conjecture in some new cases.

The loop Murnaghan–Nakayama rule

We give a combinatorial proof of a natural generalization of the Murnaghan---Nakayama rule to loop Schur functions. We also introduce shifted loop Schur functions and prove that they satisfy a similar relation.

A Murnaghan-Nakayama Rule for Generalized Demazure Atoms

We prove an analogue of the Murnaghan-Nakayama rule to express the product of a power symmetric function and a generalized Demazure atom in terms of generalized Demazure atoms.

Near-central permutation factorization and Strahovʼs generalized Murnaghan–Nakayama rule

The (p,q,n)-dipole problem is a map enumeration problem, arising in perturbative Yang–Mills theory, in which the parameters p and q, at each vertex, specify the number of edges separating of two distinguished edges. Combinatorially, it is notable for being a permutation factorization problem which does not lie in the centre of C[Sn], rendering the problem inaccessible through the character-theoretic methods often employed to study such problems. This paper gives a solution to this problem on all orientable surfaces when q=n−1, which is a combinatorially significant special case: it is a near-central problem. We give an encoding of the (p,n−1,n)-dipole problem as a product of standard basis elements in the centralizer Z1(n) of the group algebra C[Sn] with respect to the subgroup Sn−1. The generalized characters arising in the solution to the (p,n−1,n)-dipole problem are zonal spherical functions of the Gelʼfand pair (Sn×Sn−1,diag(Sn−1)) and are evaluated explicitly. This solution is used to prove that, for a given surface, the numbers of (p,n−1,n)-dipoles and (n+1−p,n−1,n)-dipoles are equal, a fact for which we have no combinatorial explanation. These techniques also give a solution to a near-central analogue of the problem of decomposing a full cycle into two factors of specified cycle type.

Skew quantum Murnaghan–Nakayama rule

We extend recent results of Assaf and McNamara on a skew Pieri rule and a skew Murnaghan---Nakayama rule to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf---McNamara's original proof, and one via Lam---Lauve---Sotille's skew Littlewood---Richardson rule. We end with some conjectures for skew rules for Hall---Littlewood polynomials.

The Murnaghan–Nakayama rule for k-Schur functions

We prove the Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse, that is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene.

The Murnaghan―Nakayama rule for k-Schur functions

We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene. Nous prouvons une règle de Murnaghan-Nakayama pour les fonctions de k-Schur de Lapointe et Morse, c'est-à-dire que nous donnons une formule explicite pour le développement du produit d'une fonction symétrique "somme de puissances'' et d'une fonction de k-Schur en termes de fonctions k-Schur. Ceci est prouvé en utilisant les fonctions non commutatives k-Schur en termes d'algèbre nilCoxeter introduite par Lam et l'analogue affine des fonctions symétriques non commutatives de Fomin et Greene.

Skew quantum Murnaghan-Nakayama rule

In this extended abstract, we extend recent results of Assaf and McNamara, the skew Pieri rule and the skew Murnaghan-Nakayama rule, to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf-McNamara's original proof, and one via Lam-Lauve-Sotille's skew Littlewood-Richardson rule. Dans cet article nous élargissons le cadre de résultats récents de Assaf et McNamara, la règle dissymétrique de Pieri et la règle dissymétrique de Murnaghan-Nakayama, pour obtenir une identité plus générale donnant un développement élégant du produit de la fonction de Schur dissymétrique par une somme de puissances quantiques, en termes de fonctions de Schur dissymétriques. Nous donnons deux démonstrations, la première suivant l'approche de Assaf-McNamara et la deuxième par le biais de la règle dissymétrique de Littlewood-Richardson obtenue par Lam-Lauve-Sotille.

Application of graph combinatorics to rational identities of type $A$

To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} - x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize identities linked to the Murnaghan-Nakayama rule, is a sum of its images by certain permutations of the variables. The sets of permutations that we consider are the linear extensions of oriented graphs. We explain how to compute this rational function, using the combinatorics of the graph $G$. We also establish a link between an algebraic property of the rational function (the factorization of the numerator) and a combinatorial property of the graph (the existence of a disconnecting chain).

Noncommutative Schur functions and their applications

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood–Richardson and Murnaghan–Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.

Schur Functions and Alternating Sums

We discuss several well known results about Schur functions that can be proved using cancellations in alternating summations; notably we shall discuss the Pieri and Murnaghan-Nakayama rules, the Jacobi-Trudi identity and its dual (Von Nägelsbach-Kostka) identity, their proofs using the correspondence with lattice paths of Gessel and Viennot, and finally the Littlewood-Richardson rule. Our our goal is to show that the mentioned statements are closely related, and can be proved using variations of the same basic technique. We also want to emphasise the central part that is played by matrices over $\{0,1\}$ and over $\Bbb N$; we show that the Littlewood-Richardson rule as generalised by Zelevinsky has elegant formulations using either type of matrix, and that in both cases it can be obtained by two successive reductions from a large signed enumeration of such matrices, where the sign depends only on the row and column sums of the matrix.

Virasoro action on Schur function expansions, skew Young tableaux, and random walks

It is known that some matrix integrals over U(n) satisfy an sl(2,R)-algebra of Virasoro constraints. Acting with these Virasoro generators on 2-dimensional Schur function expansions leads to difference relations on the coefficients of this expansions. These difference relations, set equal to zero, are precisely the backward and forward equations for non-intersecting random walks. The transition probabilities for these random walks appear as the coefficients of an expansion of U(n)-matrix integrals (of the type above), by inserting in the integral the product of two Schur polynomials associated with two partitions; the latter are specified by the initial and final positions of the non-intersecting random walk. An essential ingredient in this work is the generalization of the Murnaghan-Nakayama rule to the action of Virasoro on Schur polynomials.

A $\lambda$-ring Frobenius Characteristic for $G\wr S_n$

A $\lambda$-ring version of a Frobenius characteristic for groups of the form $G \wr S_n$ is given. Our methods provide natural analogs of classic results in the representation theory of the symmetric group. Included is a method decompose the Kronecker product of two irreducible representations of $G\wr S_n$ into its irreducible components along with generalizations of the Murnaghan-Nakayama rule, the Hall inner product, and the reproducing kernel for $G\wr S_n$.

The computational complexity of rules for the character table of Sn

The Murnaghan–Nakayama rule is the classical formula for computing the character table of S n . Y. Roichman (Adv. Math. 129 (1997) 25) has recently discovered a rule for the Kazhdan–Lusztig characters of q Hecke algebras of type A, which can also be used for the character table of S n . For each of the two rules, we give an algorithm for computing entries in the character table of S n . We then analyze the computational complexity of the two algorithms, and in the case of characters indexed by partitions in the ( k,ℓ) hook, compare their complexities to each other. It turns out that the algorithm based on the Murnaghan–Nakayama rule requires far less operations than the other algorithm. We note the algorithms’ complexities’ relation to two enumeration problems of Young diagrams and Young tableaux.

Character Formulas for q -Rook Monoid Algebras

The q-rook monoid R n(q) is a semisimple ℂ(q)-algebra that specializes when q → 1 to ℂ[R n], where R n is the monoid of n × n matrices with entries from {0, 1} and at most one nonzero entry in each row and column. We use a Schur-Weyl duality between R n(q) and the quantum general linear group $$U_q {\mathfrak{g}}{\mathfrak{l}}(r)$$ to compute a Frobenius formula, in the ring of symmetric functions, for the irreducible characters of R n(q). We then derive a recursive Murnaghan-Nakayama rule for these characters, and we use Robinson-Schensted-Knuth insertion to derive a Roichman rule for these characters. We also define a class of standard elements on which it is sufficient to compute characters. The results for R n(q) specialize when q = 1 to analogous results for R n.

Noncommutative schur functions and their applications

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood-Richardson and Murnaghan-Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.