128 research outputs found
Generalized Foulkes modules and maximal and minimal constituents of plethysms of Schur functions
No full textThis paper proves a combinatorial rule giving all maximal and minimal partitions such that the Schur function appears in a plethysm of two arbitrary Schur functions. Determining the decomposition of these plethysms has been identified by Stanley as a key open problem in algebraic combinatorics. As corollaries we prove three conjectures of Agaoka on the partitions labeling the lexicographically greatest and least Schur functions appearing in an arbitrary plethysm. We also show that the multiplicity of the Schur function labelled by the lexicographically least constituent may be arbitrarily large. The proof is carried out in the symmetric group and gives an explicit non-zero homomorphism corresponding to each maximal or minimal partition
Character deflations and a generalization of the Murnaghan--Nakayama rule
No full textGiven natural numbers m and n, we define a deflation map from the characters of the symmetric group S_{mn} to the characters of S_n. This map is obtained by first restricting a character of S_{mn} to the wreath product S_m ?S_n, and then taking the sum of the irreducible constituents of the restricted character on which the base group S_m ×?×S_m 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
William Paget and the late-Henrican polity, 1543-1547
No full textThis thesis explores the late-Henrican polity through the archive and perspective of William
Paget, Henry VIII's secretary at the end of his reign. Paget's papers as secretary (1543-1547), that
form the basis of the thesis, are an extensive, unique and relatively under-used source. From this
starting-point Paget's role as secretary is explored and he is revealed as the personal servant of
the king, whose natural environment was the court. As such he was an influential source of
counsel and perhaps the key patronage-broker at court. In this context Paget also had a significant
influence over the operation of the dry stamp at the end of the reign. Equally, Paget's role in
shaping the function of the secretary and his relations with the recently formed privy council was
of considerable importance, providing the template for later Tudor secretaries.
Diplomacy in the uncertain world of the 1540s was one of Paget's primary concerns and his
priorities can be seen as trying to provide security and stability for the realm. This is revealed not
only in his 'Consultation' of August 1546 but also in his diplomacy with the French, the
Schmalkaldic League and the Papacy. In this he sometimes found himself at odds with the king
and leading a privy council united in a desire for peace.
Politically Paget has traditionally been cast as an ambitious politique, the 'master of practices'
and part of the earl of Hertford's reform party. Whilst acknowledging Paget's close relations with
Hertford this thesis questions the factional interpretation of the last years of the reign and argues
that the predominant concern of Paget and his fellow privy councillors was a peaceful succession
in which unanimity rather than conflict was the key-note
On the structure of Foulkes modules for the symmetric group
No full textThis thesis concerns the structure of Foulkes modules for the symmetric group. We study `ordinary' Foulkes modules , where and are natural numbers, which are permutation modules arising from the action on cosets of . We also study a generalisation of these modules , labelled by a partition of , which we call generalised Foulkes modules.
Working over a field of characteristic zero, we investigate the module structure using semistandard homomorphisms. We identify several new relationships between irreducible constituents of and , where is a natural number, and also apply the theory to twisted Foulkes modules, which are labelled by , obtaining analogous results.
We make extensive use of character-theoretic techniques to study , the ordinary character afforded by the Foulkes module , and we draw conclusions about near-minimal constituents of in the case where is even. Further, we prove a recursive formula for computing character multiplicities of any generalised Foulkes character , and we decompose completely the character in the cases where has either two rows or two columns, or is a hook partition.
Finally, we examine the structure of twisted Foulkes modules in the modular setting. In particular, we answer questions about the structure of over fields of prime characteristic
Decompositions of some twisted Foulkes characters
No full textWe decompose the twisted Foulkes characters ϕ(2n)ν, or equivalently the plethysm sν∘s(2), in the cases where ν has either two rows or two columns, or is a hook partition
Minimal and maximal constituents of twisted Foulkes characters
No full textWe 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 . As a corollary we prove
two conjectures of Agaoka on the lexicographically least constituents of the plethysms and
Violet Paget (London, England) to Matilda Paget (Florence, Italy)
No full textIs dissatisfied with her book sales, I am decidedly an unsuccessful author, well known but not read
Violet Paget (London, England) to Matilda Paget (Siena, Italy)
No full textRobinsons had a dance for 70 people last night; Fischer Unwin, publisher, called, wanting advice on life of Garibaldi; feels she\u27s still obscure as an author; 23rd: Madeline and Gosse visited; went to Mary Robinson\u27s lecture on Lake Poets at Working Women\u27s College; more negotiations with Elliot Stock over Eugene\u27s book; 24th: dined at Leslie Stephen\u27s; 25th: Symonds is not well; she and Mary saw Wagner\u27s Trista
Violet Paget (between Dieppe and Newhaven) to Matilda Paget (Bagni di Lucca, Italy)
No full textArrived in Roven on the first day of the races there and left because of the crowds; boarded a boat at Dieppe and will arrive in Newhaven, England; John Sargent calls the author Rollinat a beast without any talent
Violet Paget (London, England) to Matilda Paget (Bagni di Lucca, Italy)
No full textTold Knowles of Nineteenth Century she would rather have Perigot appear in the Contemporary Review; dined with the Crawleys; went with Mabel Robinson to hear Ulster Protestants talk about Home Rule; visited Mary Wakefield, Mrs. Green and Miss Chapman, author of the New Godiva ; has elaborate travelling plans for the next month
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