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Topics in Computational Algebra

The most function of those lectures is first to in brief survey the elemental con­ nection among the illustration idea of the symmetric workforce Sn and the idea of symmetric services and moment to teach how combinatorial equipment that come up evidently within the conception of symmetric capabilities bring about effective algorithms to specific a number of prod­ ucts of representations of Sn by way of sums of irreducible representations.

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P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✵ ❜❛rs ✢♦❛t t❤r♦✉❣❤ t❤❡ ❛✐r ❛♥❞ ❧❛♥❞ ♦♥ t❤❡ t❡❛❝❤❡r✬s ❞❡s❦✳ ❆♥❞✱ ❛s q✉✐❝❦❧② ❛s s❤❡ ❛♣♣❡❛r❡❞✱ ❙❛❧❧② ✐s ❣♦♥❡ t♦ ❞♦ ♠♦r❡ ❣♦♦❞ ✐♥ t❤❡ ✇♦r❧❞✳ ▲❡t s r❡♣r❡s❡♥t t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ss✱ ❛♥❞ c r❡♣r❡s❡♥t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❛♥❞② ❜❛rs ❞✐str✐❜✉t❡❞✳ ❚✇♦ ❢♦r ❡❛❝❤ st✉❞❡♥t✱ ❛♥❞ ✜✈❡ ❢♦r t❤❡ t❡❛❝❤❡r✳ ❛✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ t♦ s❤♦✇ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ❙❛❧❧② ❣❛✈❡ ♦✉t✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts✳ c (s) =❴❴❴❴❴❴ ❜✳ ❯s❡ t❤❛t ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ t❤❡r❡ ✇❡r❡ ✷✵ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ssr♦♦♠✱ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ✇❡r❡ ❞✐str✐❜✉t❡❞❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❝✳ ◆♦✇ ✉s❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ ❙❛❧❧② ❞✐str✐❜✉t❡❞ ✸✺ ❝❛♥❞② ❜❛rs✱ ❤♦✇ ♠❛♥② st✉❞❡♥ts ✇❡r❡ ✐♥ t❤❡ ❝❧❛ss❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✵ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = ✐s ✏❙✉❜tr❛❝t t❤r❡❡✱ t❤❡♥ t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ f (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✶ ❚❤❡ ❢✉♥❝t✐♦♥ y (x) ✐s ✏●✐✈❡♥ ❛♥② ♥✉♠❜❡r✱ r❡t✉r♥ ✻✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ y (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✷ z (x) = x2 − 6x + 9 ❛✳ z (−1) =❴❴❴❴❴❴ ❜✳ z (0) = ❴❴❴❴❴❴ ❝✳ z (1) =❴❴❴❴❴❴ ❞✳ z (3) =❴❴❴❴❴❴ ❡✳ z (x + 2) =❴❴❴❴❴❴ ❢✳ z (z (x)) =❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✸ ❖❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts ♦❢ ♥✉♠❜❡rs✱ ✐♥❞✐❝❛t❡ ✇❤✐❝❤ ♦♥❡s ❝♦✉❧❞ ♣♦ss✐❜❧② ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥✳ ❆❧❧ ■ ♥❡❡❞ ✐s ❛ ✏❨❡s✑ ♦r ✏◆♦✑✖②♦✉ ❞♦♥✬t ❤❛✈❡ t♦ t❡❧❧ ♠❡ t❤❡ ❢✉♥❝t✐♦♥✦ ✭❇✉t ❣♦ ❛❤❡❛❞ ❛♥❞ ❞♦✱ ✐❢ ②♦✉ ✇❛♥t t♦.

P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✵ ❜❛rs ✢♦❛t t❤r♦✉❣❤ t❤❡ ❛✐r ❛♥❞ ❧❛♥❞ ♦♥ t❤❡ t❡❛❝❤❡r✬s ❞❡s❦✳ ❆♥❞✱ ❛s q✉✐❝❦❧② ❛s s❤❡ ❛♣♣❡❛r❡❞✱ ❙❛❧❧② ✐s ❣♦♥❡ t♦ ❞♦ ♠♦r❡ ❣♦♦❞ ✐♥ t❤❡ ✇♦r❧❞✳ ▲❡t s r❡♣r❡s❡♥t t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ss✱ ❛♥❞ c r❡♣r❡s❡♥t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❛♥❞② ❜❛rs ❞✐str✐❜✉t❡❞✳ ❚✇♦ ❢♦r ❡❛❝❤ st✉❞❡♥t✱ ❛♥❞ ✜✈❡ ❢♦r t❤❡ t❡❛❝❤❡r✳ ❛✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ t♦ s❤♦✇ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ❙❛❧❧② ❣❛✈❡ ♦✉t✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts✳ c (s) =❴❴❴❴❴❴ ❜✳ ❯s❡ t❤❛t ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ t❤❡r❡ ✇❡r❡ ✷✵ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ssr♦♦♠✱ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ✇❡r❡ ❞✐str✐❜✉t❡❞❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❝✳ ◆♦✇ ✉s❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ ❙❛❧❧② ❞✐str✐❜✉t❡❞ ✸✺ ❝❛♥❞② ❜❛rs✱ ❤♦✇ ♠❛♥② st✉❞❡♥ts ✇❡r❡ ✐♥ t❤❡ ❝❧❛ss❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✵ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = ✐s ✏❙✉❜tr❛❝t t❤r❡❡✱ t❤❡♥ t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ f (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✶ ❚❤❡ ❢✉♥❝t✐♦♥ y (x) ✐s ✏●✐✈❡♥ ❛♥② ♥✉♠❜❡r✱ r❡t✉r♥ ✻✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ y (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✷ z (x) = x2 − 6x + 9 ❛✳ z (−1) =❴❴❴❴❴❴ ❜✳ z (0) = ❴❴❴❴❴❴ ❝✳ z (1) =❴❴❴❴❴❴ ❞✳ z (3) =❴❴❴❴❴❴ ❡✳ z (x + 2) =❴❴❴❴❴❴ ❢✳ z (z (x)) =❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✸ ❖❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts ♦❢ ♥✉♠❜❡rs✱ ✐♥❞✐❝❛t❡ ✇❤✐❝❤ ♦♥❡s ❝♦✉❧❞ ♣♦ss✐❜❧② ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥✳ ❆❧❧ ■ ♥❡❡❞ ✐s ❛ ✏❨❡s✑ ♦r ✏◆♦✑✖②♦✉ ❞♦♥✬t ❤❛✈❡ t♦ t❡❧❧ ♠❡ t❤❡ ❢✉♥❝t✐♦♥✦ ✭❇✉t ❣♦ ❛❤❡❛❞ ❛♥❞ ❞♦✱ ✐❢ ②♦✉ ✇❛♥t t♦.

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