Murnaghan-Nakayama Rule Research Articles

Proof of the Plethystic Murnaghan–Nakayama Rule Using Loehr’s Labelled Abacus

AbstractThe plethystic Murnaghan–Nakayama rule describes how to decompose the product of a Schur function and a plethysm of the form $$p_r\circ h_m$$ p r ∘ h m as a sum of Schur functions. We provide a short, entirely combinatorial proof of this rule using the labelled abaci introduced in Loehr (SIAM J Discrete Math 24(4):1356–1370, 2010).

A Spin Analog of the Plethystic Murnaghan–Nakayama Rule

A Spin Analog of the Plethystic Murnaghan–Nakayama Rule

Biharmonic conjectures on hypersurfaces in a space form

We apply the Murnaghan-Nakayama rule in the representation theory of symmetric groups to develop new techniques for studying biharmonic hypersurfaces in a space form. As applications of the new techniques, we settle the well-known Chen’s conjecture on biharmonic hypersurfaces in R 6 \Bbb R^6 and BMO conjecture on biharmonic hypersurfaces in S 6 \mathbb S^6 .

A Combinatorial Proof of a Symmetric Group Character Involution

We give a short combinatorial proof based on the Murnaghan-Nakayama rule of the symmetric group character identity , where is the conjugate of the partition .

A Murnaghan–Nakayama Rule for Grothendieck Polynomials of Grassmannian Type

A Murnaghan–Nakayama Rule for Grothendieck Polynomials of Grassmannian Type

A Generalization of the Murnaghan–Nakayama Rule for K-k-Schur and k-Schur Functions

Abstract The $K$-$k$-Schur functions and $k$-Schur functions appeared in the study of $K$-theoretic and affine Schubert Calculus as polynomial representatives of Schubert classes. In this paper, we introduce a new family of symmetric functions $\mathcal {F}_{\lambda }^{(k)}$, that generalizes the constructions via the Pieri rule of $K$-$k$-Schur functions and $ k$-Schur functions. Then we obtain the Murnaghan–Nakayama rule for the generalized functions. The rule is described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases and connections promise to be detailed in the future.

Murnaghan–Nakayama Rule and Spin Bitrace for the Hecke–Clifford Algebra

AbstractA Pfaffian-type Murnaghan–Nakayama rule is derived for the Hecke–Clifford algebra $\mathcal{H}^{c}_{n}$ based on the Frobenius formula and vertex operators, and this leads to a combinatorial version via the tableaux realization of Schur’s $Q$-functions. As a consequence, a general formula for the irreducible characters $\zeta ^{\lambda }_{\mu }(q)$ using partition-valued functions is derived. Meanwhile, an iterative formula on the indexing partition $\lambda $ via the Pieri rule is also deduced. As applications, some compact formulae of the irreducible characters are given for special partitions and a symmetric property of the irreducible character is found. We also introduce the spin bitrace as the analogue of the bitrace for the Hecke algebra and derive its general combinatorial formula. Tables of irreducible characters are listed for $n\leq 7.$

A refinement of the Murnaghan-Nakayama rule by descents for border strip tableaux

Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer (1974) and James & Kerber (1984) imply that, mysteriously, its evaluation at a \(k\)-th primitive root of unity yields the number of border strip tableaux with all strips of size \(k\), up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for evaluating an irreducible character of the symmetric group at a rectangular partition. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we introduce a new statistic for border strip tableaux, extending the classical definition of descents in standard Young tableaux. Curiously, it turns out that our new statistic is very closely related to a descent set for tuples of standard Young tableaux appearing in the quasisymmetric expansion of LLT polynomials given by Haglund, Haiman and Loehr (2005).Mathematics Subject Classifications: 05A19, 05E10Keywords: Border strip tableaux, descents, Murnaghan-Nakayama rule, fake degree

On characters of wreath products

A character identity which relates irreducible character values of the hyperoctahedral group \(B_n\) to those of the symmetric group \(S_{2n}\) was recently proved by Lübeck and Prasad. Their proof is algebraic and involves Lie theory. We present a short combinatorial proof of this identity, as well as a generalization to other wreath products.Mathematics Subject Classifications: 20C30, 20E22, 05E10Keywords: Character identity, wreath product, partition, Murnaghan-Nakayama rule, colored permutations

On irreducible characters of the Iwahori-Hecke algebra in type A

In this paper, we use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type A. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. We also formulate a determinant type Murnaghan-Nakayama formula and give another proof of the combinatorial Murnaghan-Nakayama rule. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Leduc-Ram formula.

A Murnaghan Nakayama rule for schur functions in superspace

A Murnaghan Nakayama rule for schur functions in superspace

Combinatorial description of the cohomology of the affine flag variety

We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We introduce Murnaghan-Nakayama elements and Dunklelements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutativealgebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, weobtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to expressk-Schur functions in terms of power sum symmetric functions. We also provide the defi-nition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of theaffine flag variety.

A Proof of the Murnaghan–Nakayama Rule Using Specht Modules and Tableau Combinatorics

The Murnaghan–Nakayama rule is a combinatorial rule for the character values of symmetric groups. We give a new combinatorial proof by explicitly finding the trace of the representing matrices in the standard basis of Specht modules. This gives an essentially bijective proof of the rule. A key lemma is an extension of a straightening result proved by the second author to skew tableaux. Our module theoretic methods also give short proofs of Pieri’s rule and Young’s rule.

P-Partitions and Quasisymmetric Power Sums

Abstract The $(P, \omega )$-partition generating function of a labeled poset $(P, \omega )$ is a quasisymmetric function enumerating certain order-preserving maps from $P$ to ${\mathbb{Z}}^+$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $\{\psi _{\alpha }\}$. Using this expansion, we show that connected, naturally labeled posets have irreducible $P$-partition generating functions. We also show that series-parallel posets are uniquely determined by their partition generating functions. We conclude by giving a combinatorial interpretation for the coefficients of the $\psi _{\alpha }$-expansion of the $(P, \omega )$-partition generating function akin to the Murnaghan–Nakayama rule.

Combinatorial description of the cohomology of the affine flag variety

We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We introduce Murnaghan-Nakayama elements and Dunklelements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutativealgebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, weobtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to expressk-Schur functions in terms of power sum symmetric functions. We also provide the defi-nition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of theaffine flag variety.

Two Murnaghan-Nakayama Rules in Schubert Calculus

The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan- Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming fromthe Chern character. Like the classical rule, both rules are multiplicity-free signed sums.

Superposition induced topology changes in quantum gravity

We show that superpositions of classical states in quantum gravity with fixed topology can lead to new classical states with a different topology. We study this phenomenon in a particular limit of the LLM geometries. In this limit, the UV complete minisuperspace of allowed quantum states is exactly given by the Hilbert space of a free chiral boson in two dimensions. We construct this chiral boson purely in terms of combinatorial objects associated with the permutation group. As a byproduct of this analysis, we rederive the Murnaghan-Nakayama rule for characters of the permutation group. We are able to express this rule in terms of operator relations for raising and lowering operators on the Hilbert space of states in a free fermion basis. Our construction provides a preferred notion of bulk locality by studying an appropriate notion of D-brane state generating functions. We describe how multi-droplet LLM geometries with different topologies give new classical limits of the free chiral boson, even though they can be written as superpositions of coherent states with trivial topology. As a consequence, topology cannot be accessed by a single operator measurement in this quantum system. We study other non-linear measurements in the quantum wave-function, based on uncertainty and entanglement between modes of the chiral boson, that can be used as order parameters to measure the topology of such states.

Multiplicative structures of the immaculate basis of non-commutative symmetric functions

We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the basis of Schur functions for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutative versions of the Littlewood–Richardson rule and the Murnaghan–Nakayama rule. A surprising relation develops among non-commutative Littlewood–Richardson coefficients, which has implications to the commutative case. Finally, we interpret these new coefficients geometrically as the number of integer points inside a certain polytope.

Perfect isometries and Murnaghan-Nakayama rules

Acknowledgement The authors wish to thank the referee for a very careful and precise reading of several earlier versions of this manuscript. They are grateful for the number, quality and helpfulness of comments and suggestions received.

Corrections to: “A Murnaghan–Nakayama rule for values of unipotent characters in classical groups”

Corrections to: “A Murnaghan–Nakayama rule for values of unipotent characters in classical groups”