For a finite real reflection group W and a W -orbit O of flats in its reflection arrangement - or equivalently a conjugacy class of its parabolic subgroups - the authors introduce a statistic noninv O (w) on w in W that counts the number of O -noninversions of w . This generalises the classical (non-)inversion statistic for permutations w in the symmetric group S n. The authors then study the operator ? O of right-multiplication within the group algebra CW by the element that has noninv O (w) as its coefficient on w.
Each year since 1996 the universities of Bergen, Oslo and Trondheim have organized summer schools in Nordfjordeid in various topics in algebra and related ?elds. Nordfjordeid is the birthplace of Sophus Lie, and is a village on the western coast of Norway situated among fjords and mountains, with sp- tacularscenerywhereveryougo. AssuchitisawelcomeplaceforbothNor- gian and international participants and lecturers. The theme for the summer school in 2003 was Algebraic Combinatorics. The organizing committee c- sisted of Gunnar Fløystad and Stein Arild Strømme (Bergen), Geir Ellingsrud and Kristian Ranestad (Oslo), and Alexej Rudakov and Sverre Smalø (Tro- heim). The summer school was partly ?nanced by NorFa-Nordisk Forsker- danningsakademi. With combinatorics reaching into and playing an important part of ever more areas in mathematics, in particular algebra, algebraic combinatorics was a timely theme. The ?st lecture series “Hyperplane arrangements” was given by Peter Orlik. He came as a refugee to Norway, eighteen years old, after the insurrection in Hungary in 1956. Despite now having lived more than four decades in the United States, he impressed us by speaking ?uent Norwegian without a trace of accent. The second lecture series “Discrete Morse theory and free resolutions” was given by Volkmar Welker. These two topics ori- nate back in the second half of the nineteenth century with simple problems on arrangements of lines in the plane and Hilberts syzygy theorem.
Each year since 1996 the universities of Bergen, Oslo and Trondheim have organized summer schools in Nordfjordeid in various topics in algebra and related ?elds. Nordfjordeid is the birthplace of Sophus Lie, and is a village on the western coast of Norway situated among fjords and mountains, with sp- tacularscenerywhereveryougo. AssuchitisawelcomeplaceforbothNor- gian and international participants and lecturers. The theme for the summer school in 2003 was Algebraic Combinatorics. The organizing committee c- sisted of Gunnar Fløystad and Stein Arild Strømme (Bergen), Geir Ellingsrud and Kristian Ranestad (Oslo), and Alexej Rudakov and Sverre Smalø (Tro- heim). The summer school was partly ?nanced by NorFa-Nordisk Forsker- danningsakademi. With combinatorics reaching into and playing an important part of ever more areas in mathematics, in particular algebra, algebraic combinatorics was a timely theme. The ?st lecture series “Hyperplane arrangements” was given by Peter Orlik. He came as a refugee to Norway, eighteen years old, after the insurrection in Hungary in 1956. Despite now having lived more than four decades in the United States, he impressed us by speaking ?uent Norwegian without a trace of accent. The second lecture series “Discrete Morse theory and free resolutions” was given by Volkmar Welker. These two topics ori- nate back in the second half of the nineteenth century with simple problems on arrangements of lines in the plane and Hilberts syzygy theorem.
For a finite real reflection group W and a W -orbit O of flats in its reflection arrangement - or equivalently a conjugacy class of its parabolic subgroups - the authors introduce a statistic noninv O (w) on w in W that counts the number of O -noninversions of w . This generalises the classical (non-)inversion statistic for permutations w in the symmetric group S n. The authors then study the operator ? O of right-multiplication within the group algebra CW by the element that has noninv O (w) as its coefficient on w.
In a series of papers Tsirelson constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So the author connects each continuous tensor product system of Hilbert spaces with measure types of distributions of random (closed) sets in $[0,1]$ or $\mathbb R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the Tsirelson examples, that the classification scheme for product systems into types $\mathrm{I}_n$, $\mathrm{II}_n$ and $\mathrm{III}$ is not complete. Moreover, based on a detailed study of this kind of measure types, the author constructs for each stationary factorising measure type a continuous tensor product system of Hilbert spaces such that this measure type arises as the before mentioned invariant.
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