##### Department of Mathematics,

University of California San Diego

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### Lie Groups Seminar

## Bertram Kostant

#### MIT

## Experimental evidence for the occurrence of E(8) in nature and the radii of the Gossett circles

##### Abstract:

A recent experimental discovery involving the spin structure of electrons in a cold one dimensional magnet points to a model involving the exceptional Lie group E(8). The model predicts 8 particles the ratio of whose masses are the same as the ratios of the radii of the circles in the famous Gossett diagram (going back to 1900) of what is now understood to be a 2 dimensional projection of the 240 roots of E(8) arranged in 8 concentric circles. The ratio of the radii of the two smallest circles (read 2 smallest masses) is the golden number. This beautifully has been found experimentally. The ratio of the radii of the other masses has been written down conjecturally by Zamolodchikov. This again agrees with the analogous statement for the radii of the Gossett circles. Some time ago we found an operator A (very easily defined and reexpressed by Vogan as an element of the group algebra of the Weyl group) on 8-space whose spectrum is exactly the squares of the radii of the Gossett circles. The operator A is written in terms of the coefficients $n_i$ of the highest root. In McKay theory the $n_i$ are the dimensions of the irreducible representations of the binary icosahedral group. Our result works for any simple Lie group not just E(8).

Host: Nolan Wallach

### February 23, 2010

### 2:00 PM

### AP&M 6402

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