Irreducible Constituents Research Articles

Criticality as a Determinant of Integrated Information Φ in Human Brain Networks

Integrated information theory (IIT) describes consciousness as information integrated across highly differentiated but irreducible constituent parts in a system. However, in a complex dynamic system such as the brain, the optimal conditions for large integrated information systems have not been elucidated. In this study, we hypothesized that network criticality, a balanced state between a large variation in functional network configuration and a large constraint on structural network configuration, may be the basis of the emergence of a large , a surrogate of integrated information. We also hypothesized that as consciousness diminishes, the brain loses network criticality and decreases. We tested these hypotheses with a large-scale brain network model and high-density electroencephalography (EEG) acquired during various levels of human consciousness under general anesthesia. In the modeling study, maximal criticality coincided with maximal . The EEG study demonstrated an explicit relationship between , criticality, and level of consciousness. The conscious resting state showed the largest and criticality, whereas the balance between variation and constraint in the brain network broke down as the response rate dwindled. The results suggest network criticality as a necessary condition of a large in the human brain.

Construction of the irreducible modular representations of a finite group

A complete procedure is described for constructing the irreducible KG-modules and their Brauer characters, where K is a finite field of characteristic p and G is a finite permutation or matrix group. The central idea is to construct a sequence {S1,…,Sn} of KG-modules, each having relatively small dimension, such that each Si has one or more irreducible constituents that are not constituents of S1,…,Si−1. The Meataxe, used in conjunction with condensation, is used to extract the new irreducibles from each Si. The algorithm has been implemented in Magma and is capable of constructing irreducibles of dimension over 200000.

Irreducible constituents of products of characters

In this article, we give some results about the relationship between the structure of a finite solvable group G and the kernels of irreducible constituents of the character where χ is an irreducible character of G. For example, we obtain some results for the inequality which is known as the Taketa inequality, where dl(G) is the derived length of G and is the number of distinct irreducible complex character degrees of G.

Zeta functions associated to admissible representations of compact 𝑝-adic Lie groups

Let G G be a profinite group. A strongly admissible smooth representation ϱ \varrho of G G over C {\mathbb {C}} decomposes as a direct sum ϱ ≅ ⨁ π ∈ Irr ( G ) m π ( ϱ ) π \varrho \cong \bigoplus _{\pi \in \text {Irr}(G)} m_\pi (\varrho ) \, \pi of irreducible representations with finite multiplicities m π ( ϱ ) m_\pi (\varrho ) such that, for every positive integer n n , the number r n ( ϱ ) r_n(\varrho ) of irreducible constituents of dimension n n is finite. Examples arise naturally in the representation theory of reductive groups over nonarchimedean local fields. In this article we initiate an investigation of the Dirichlet generating function \[ ζ ϱ ( s ) = ∑ n = 1 ∞ r n ( ϱ ) n − s = ∑ π ∈ Irr ( G ) m π ( ϱ ) ( dim ⁡ π ) s \zeta _\varrho (s) = \sum \nolimits _{n=1}^\infty r_n(\varrho ) n^{-s} = \sum \nolimits _{\pi \in \text {Irr}(G)} \frac {m_\pi (\varrho )}{(\dim \pi )^s} \] associated to such a representation ϱ \varrho . Our primary focus is on representations ϱ = Ind H G ( σ ) \varrho = \text {Ind}_H^G(\sigma ) of compact p p -adic Lie groups G G that arise from finite-dimensional representations σ \sigma of closed subgroups H H via the induction functor. In addition to a series of foundational results—including a description in terms of p p -adic integrals—we establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro- p p groups. A key ingredient of our proof is Hironaka’s resolution of singularities, which yields formulae of Denef type for the relevant zeta functions. In some detail we consider representations of open compact subgroups of reductive p p -adic groups that are induced from parabolic subgroups. Explicit computations are carried out by means of complementing techniques: (i) geometric methods that are applicable via distance-transitive actions on spherically homogeneous rooted trees, and (ii) the p p -adic Kirillov orbit method. Approach (i) is closely related to the notion of Gelfand pairs and works equally well in positive defining characteristic.

PROJECTIVE CHARACTERS WITH PRIME POWER DEGREES

We consider the relationship between structural information of a finite group $G$ and $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$, the set of all irreducible projective character degrees of $G$ with factor set $\unicode[STIX]{x1D6FC}$. We show that for nontrivial $\unicode[STIX]{x1D6FC}$, if all numbers in $\text{cd}_{\unicode[STIX]{x1D6FC}}(G)$ are prime powers, then $G$ is solvable. Our result is proved by classical character theory using the bijection between irreducible projective representations and irreducible constituents of induced representations in its representation group.

On permutation characters and Sylow p-subgroups of [formula omitted

Let p be an odd prime and let n be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group Sn on the cosets of a Sylow p-subgroup Pn. As a consequence, we determine the number of irreducible representations of the corresponding Hecke algebra H(Sn,Pn,1Pn).

A note on skew characters of symmetric groups

In previous work Regev used part of the representation theory of Lie superalgebras to compute the values of a character of the symmetric group whose decomposition into irreducible constituents is described by semistandard $(k,\ell)$-tableaux. In this short note we give a new proof of Regev's result using skew characters.

Induced characters of equal degree constituents

ABSTRACTIf H is a subgroup of G, we investigate the situation where the irreducible constituents of αG all have equal degree for each nonprincipal irreducible character α of H. When this situation occurs, either H is contained in G′ or G′ is contained in H. When HG, the normal closure of H in G, is proper in G′, then H is solvable, although G need not be solvable. We show that G is solvable when H < G′ and . It has been conjectured that H is subnormal in G. We present a partial result.

Graham Priest's Mathematical Analysis of the Concept of Emptiness

In his article ‘The Structure of Emptiness’ (cf. Priest 2009) Graham Priest examines the concept of emptiness in the Mādhyamaka school of Nāgārjuna and his commentators Candrakīırti and Tsongkhapa from a mathematical point of view. The approach attempted in this article does not involve any commitment to Priest's more controversial dialethic Mādhyamaka interpretation. The purpose of the present paper is to explain Priest's sketchy but very insightful interpretation of objects as non-well-founded sets in greater detail. Some problems concerning his idea to model the Mādhyamaka claim of the emptiness of emptiness by means of this kind of framework will be noted. Moreover, we will also discuss the possibility to represent the Mādhyamika's denial of the existence of irreducible constituents of empirical reality within a well-founded system of set theory. Finally, some slight mistakes in Priest's mathematical construction need to be pointed out.

Derived length when a product of characters has two nonprincipal irreducible constituents

We study the situation where a solvable group [Formula: see text] has a faithful irreducible character [Formula: see text] such that [Formula: see text] has exactly two distinct nonprincipal irreducible constituents. We prove that [Formula: see text] has derived length bounded above by 8, and provide an example of such a group having derived length 8. In particular, this improves upon a result of Adan-Bante.

On the U-module structure of the unipotent Specht modules of finite general linear groups

Let q be a prime power, G=GLn(q) and let U⩽G be the subgroup of (lower) unitriangular matrices in G. For a partition λ of n denote the corresponding unipotent Specht module over the complex field C for G by Sλ. It is conjectured that for c∈Z⩾0 the number of irreducible constituents of dimension qc of the restriction ResUG(Sλ) of Sλ to U is a polynomial in q with integer coefficients depending only on c and λ, not on q. In the special case of the partition λ=(1n) this implies a longstanding (still open) conjecture of Higman, stating that the number of conjugacy classes of U should be a polynomial in q with integer coefficients depending only on n not on q. In this paper we prove the conjecture for the case that λ=(n−m,m)(0⩽m⩽n/2) is a 2-part partition. As a consequence, we obtain a new representation theoretic construction of the standard basis of Sλ (over fields of characteristic coprime to q) defined by M. Brandt, R. Dipper, G. James and S. Lyle and an explanation of the rank polynomials appearing there.

Standard Majorana representations of the symmetric groups

Let G be a finite group, W be a $${\mathbb {R}}[G]$$R[G]-module equipped with a G-invariant positive definite bilinear form $$(\,,\,)_W$$(,)W, and X a finite generating set of W such that X is transitively permuted by G. We show a new method for computing the dimensions of the irreducible constituents of W. Further, we apply this method to Majorana representations of the symmetric groups and prove that the symmetric group $$S_n$$Sn has a Majorana representation in which every permutation of type (2, 2) of $$S_n$$Sn corresponds to a Majorana axis if and only if $$n\le 12$$n≤12.

Minimal and maximal constituents of twisted Foulkes characters

We prove combinatorial rules that give the minimal and maximal partitions labelling the irreducible constituents of a family of characters for the symmetric group that generalize Foulkes permutation characters. Restated in the language of symmetric functions, our results determine all minimal and maximal partitions that label Schur functions appearing in the plethysms s_\nu \circ s_(m). As a corollary we prove two conjectures of Agaoka on the lexicographically least constituents of the plethysms s_\nu \circ s_(m) and s_\nu \circ s_(1^m).

Derived functor modules, dual pairs and [formula omitted]-actions

Derived functors (or Zuckerman functors) play a very important role in the study of unitary representations of real reductive groups. These functors are usually applied on highest weight modules in the so-called good range and the theory is well-understood. On the other hand, there were several studies on the irreducibility and unitarizability, in which derived functors are applied to singular modules. See Enright et al. (Acta Math. 1985), for example. In this article, we apply derived functors to certain modules arising from the formalism of local theta lifting, and investigate the irreducible constituent of resulting modules. The key technique is to understand U(g)K-actions in the setting of a see-saw pair. Our results suggest that derived functor constructions are compatible with local theta liftings.

The 2A-Majorana representations of the Harada-Norton group

We show that all 2 A -Majorana representations of the Harada-Norton group F 5 have the same shape. If R is such a representation, we determine, using the theory of association schemes, the dimension and the irreducible constituents of the linear span U of the Majorana axes. Finally, we prove that, if R is based on the (unique) embedding of F 5 in the Monster, U is closed under the algebra product.

Constructible characters and ${\boldsymbol{b}}$-invariant

If $W$ is a finite Coxeter group and $\varphi$ is a weight function, Lusztig has defined {\it $\\varphi$-constructible characters} of $W$, as well as a partition of the set of irreducible characters of $W$ into {\it Lusztig $\varphi$-families}. We prove that every Lusztig $\varphi$-family contains a unique character with minimal $b$-invariant, and that every $\varphi$-constructible character has a unique irreducible constituent with minimal $b$-invariant. This generalizes Lusztig's result about {\it special characters} to the case where $\varphi$ is not constant. This is compatible with some conjectures of Rouquier and the author about {\it Calogero-Moser families} and {\it Calogero-Moser cellular characters}.

Clifford theory of characters in induced blocks

We present a new criterion to predict if a character of a finite group extends. Let G be a finite group and p a prime. For N ⊳G, we consider p-blocks b and b of N and NN (D), respectively, with (b ) = b, where D is a defect group of b. Under the assumption that G coincides with a normal subgroup G[b] of G, which was introduced by Dade early 1970’s, we give a character correspondence between the sets of all irreducible constituents of φ and those of (φ′)NG(D) where φ and φ are irreducible Brauer characters in b and b, respectively. This implies a sort of generalization of the theorem of Harris-Knorr. An important tool is the existence of certain extensions that also helps in checking the inductive Alperin-McKay and inductive Blockwise Alperin Weight conditions, due to the second author.

Restricting unipotent characters in special orthogonal groups

For all prime powers $q$ we restrict the unipotent characters of the special orthogonal groups $\text{SO}_{5}(q)$ and $\text{SO}_{7}(q)$ to a maximal parabolic subgroup. We determine all irreducible constituents of these restrictions for $\text{SO}_{5}(q)$ and a large part of the irreducible constituents for $\text{SO}_{7}(q)$.

A Note on a Bound of Adan-Bante

Let G be a finite solvable group and let χ be a nonlinear irreducible (complex) character of G. Also let $$ \eta $$ (χ) be the number of nonprincipal irreducible constituents of $$ \upchi \overline{\upchi} $$ , where $$ \overline{\upchi} $$ denotes the complex conjugate of χ. Adan-Bante proved that there exist constants C and D such that dl (G/ ker χ) ≤ C $$ \eta $$ (χ) +D. In the present work, we establish a bound lower than the Adan-Bante bound for $$ \eta $$ (χ) > 2

Irreducible constituents of minimal degree in supercharacters of the finite unitriangular groups

Let q be a prime power and U the group of lower unitriangular matrices of order n for some natural number n. We give a lower bound for the degrees of irreducible constituents of André–Yan supercharacters and classify the supercharacters having constituents whose degree assume this lower bound. Moreover we show that the number of distinct irreducible characters of U meeting this condition is a polynomial in (q−1) with nonnegative integral coefficients and exhibit monomial sources for those.